DIFFEEENTIAL EQUATIONS. A TREATISE ON FFERENTIAL EQUATIONS. BY GEORGE BOOLE, F.R.S. •ROFESSOB OF MATHEMATICS IN THE QUEEn's UNIVEESITY, IBELAND, HONOKABY MEMBEE OF THE CAMBBIDGB PHILOSOPHICAL SOCIETY. FOURTH EDITION. BOSTON COLLEGE LIBRARX CHESTNUT HILL, MASS. Honiron : MACMILLAN AND CO. 1877. \All Rights reserved. 1 dJambntigc : TEINTED ET C. J. CLAY, M.A. AT THE UNIYEKSITY PRESS. tieCi V PEEFACE. [AVE endeavoured, in the following Treatise, to convey plete an account of the present state of knowledge on Dject of Differential Equations, as was consistent with a of a work intended, primarily, for elemoDtary instruc- [t was my object, first of all, to meet the wants of those whoaad no previous acquaintance with the subject, but I also cesirednot quite to disappoint others who might seek for more advanced information. These distinct, but not inconsistent ai '1=* determined the plan of composition. The earlier sections of ch chapter contain that kind of matter which has usually bees thought suitable for the beginner, while the latter ones are devoted either to an account of recent discovery, or to the discussion of such deeper questions of principle as are likely to present themselves to the reflective student in connexion with the methods and processes of his previous course. An appen- dix to the table of contents will shew what portions of the work are regarded as sufficient for the less complete, but still not unconnected study of the subject. The principles which I have kept in view in carrying out the above design, are the following : 1st, In the exposition of methods I have adhered as closely as possible to the historical order of their development. I presume that few who have paid any attention to the history of the Mathematical Analysis, will doubt that it has been developed in a certain order, or that that order has been, to a great extent, necessary — being determined, either by steps of logical deduction, or by the successive introduction of new ideas and conceptions, when the time for their evolution had VI PEEFACE. arrived. And these are causes wliich operate in perfect mony. Each new scientific conception gives occasion to applications of deductive reasoning; but those applicai may be only possible through the methods and the proc( which belong to an earlier stage. Thus, to take an illustration from the subject of the fol ing work, — the solution of ordinary simultaneous differei ■ lai equations properly precedes that of linear partial dift^ equations of the 'first order"; and this, again, properly p- that of partial differential equations of thr fi v^u order '^ ■ ' not linear. And in this natural order were the theorie these subjects developed. Again, there exist large and ^ important classes of differential equations the solution of wl depends on somfe process of successive reduction. Now s reduction seems to have been effected at first by a repea change of variables ; afterwards, and with greater general by a combination of such transformations with others inv( ing differentiation ; last of all, and with greatest generality, symbolical methods. I think it necessary to direct attent to instances like these, because the indications which tl afford appear to me to have been, in some works of gr ability, overlooked, and b^^cause I wish to explain my motr . for departing from the precedent thus set. Now there is this reason for grounding the order of ex] sition upon the historical sequence of discovery, that by doing we are most likely to present each new form of truth the mind, precisely at that stage at which the mind is mc ' fitted to receive it, or even, like that of the discoverer, to forth to meet it. 0f the many forms of false culture, a pi mature converse with' abstractions is perhaps the most like;> to prove fatal to the growth df a masculine vigour of intelle( In accordance with the above principles I have reserv( the exposition, and, with one unimportant exception, the a j:^iieation of symbolical methods to the end of the work. Tl PEEFACE. vil propriety of this course appears to me to be confirmed by an examination of the actual processes to which symbolical methods, as applied to differential equations, lead. Generally speaking, these methods present the solution of the proposed equation as dependent upon the performance of certain inverse operations. I have endeavoured to shew in Chap, xvi., that the expressions by which these inverse operations are sym- bolized are in r^eality a species of interrogations, admitting of answers, legitimate, but differing in species and character ac- cording to the nature of the transformations to which the expressions from which they are derived have been subjected. The solutions thus obtained may be particular or general, — they may be defective, wholly or partially, or complete or redundant, in those elements of a solution which are termed arbitrary. If defective, the question arises how the defect is to be supplied ; if redundant, the more difficult question whether the redundancy is real or apparent, and in either case how it is to be dealt with, must be considered. And here the necessity of some prior acquaintance with the things themselves, rather than with the symbolic forms of their ex- pression, must become apparent. The most accomplished in the use of symbols must sometimes throw aside his abstrac- tions and resort to homelier methods for trial and verification — not doubting, in so doing, the truth which lies at the bottom of his symbolism, but distrusting his own powers. The question of the true value and proper place of sym- bolical methods is undoubtedly of great importance. Their convenient simplicity — their condensed power — must ever constitute their first claim upon attention. I believe how- ever that, in order to form a just estimate, we must consider them in another aspect, viz. as in some sort the visible mani- festation of truths relating to the intimate and vital con- nexion of language with thought — truths of which it may be presumed that we do not yet see the entire scheme and con- Vlll PREFACE. nexion. But, wliile this consideration vindicates to them a high position, it seems to me clearly to define that position. As discussions about words can never remove the difficulties that exist in things, so no skill in the use of those aids to thought which language furnishes can relieve us from the necessity of a prior and more direct study of the things which are the subjects of our reasonings. And the more exact, and the more complete, that study of things has been, the more likely shall we be to employ with advantage all instru- mental aids and appliances. But although I have, for the reasons above mentioned, treated of symbolical methods only in the latter chapters of the work, I trust that the exposition of them which is there given will repay the attention of the student. I have endea- voured to supply what appeared to me to be serious defects in their logic, and I have collected under them a large number of equations, nearly all of which are important, — from their connexion with physical science or for other reasons. 2ndly, I have endeavoured, more perhaps than it has been usual to do, to found the methods of solution of differential equations upon the study of the modes of their formation. In principle, this course is justified by a consideration of the real nature ^of inverse processes, the laws of which must be ulti- mately derived from those of the direct processes to which they stand related ; in point of expediency it is recommended by the greater simplicity, and even in some instances by the greater generality, of the demonstrations to which it leads. I would refer particularly to the demonstration of Monge's method for the solution of partial differential equations of the second order given in Chap. xv. With respect to the sources from which information has been drawn, it is proper to mention that, on questions re- lating to the theory of differential equations, my obligations are greatest to Lagrange, Jacobi, Cauchy, and, of living PEEFACE. ix writers, to Professor De Morgan. For methods and exam- ples, a very large number of memoirs English and foreign have been consulted : these are, for the most part, acknow- ledged. At the same time it is right to add that, in almost every part of the work, I found it necessary to engage more or less in original investigation, and especially in those parts which relate to Riccati's equation, to integrating factors, to singular solutions, to the inverse problems of Geometry and Optics, to partial differential equations both of the first and second order, and, as has already been intimated, to symboli- cal methods. The demonstrations scattered through the work are also many of them new, at least in form. In recent years much light has been thrown on certain classes of differential equations by the researches of Jacobi on the Calculus of Variations, and of the same great analyst, with Sir W. R. Hamilton and others, on Theoretical Djma- mics. I have thought it more accordant with the design of an elementary treatise to endeavour to prepare the way for this order of inquiries than to enter systematically upon them. This object has been kept in view in the writing of various portions of the following work, and more particularly of that which relates to partial differential equations of the first order. GEORGE BOOLE. Queen's College, Cobk, February, 1859. PREFACE TO THE SECOND EDITION. In composing his Treatise on Differential Equations Pro- fessor Boole found himself deeply interested in the subject to which his first labours as an original investigator had been devoted. In consequence he determined soon after the publication of the volume to continue his studies and re- searches with the design of ultimately reconstructing the Treatise on a more extensive scale. During the last six years of his life he worked steadily at this object; and he was about to send the first sheets of the new edition to the press when he was attacked by the illness which terminated in his sudden and lamented death. His manuscripts were entrusted to me early in the present year. After careful consideration it seemed to me that the best plan to pursue was to reprint the original volume, and to collect into a supplementary volume the additional matter which had been prepared for enlarging the work. The pro- priety, I might ahnost say the necessity, of this course will be shewn more conveniently in the preface to the supple- mentary volume, which will soon be published. The present volume then is a reprint of the original Treatise with changes and corrections, some of which were indicated in Professor Boole's interleaved copy, and some of which have been made on my own authority. The sheets have been carefully read by the Kev. J. Sephton, Fellow of St John's College, as well as by myself; and I trust that few misprints or errors will now be found in the volume. I. TODHUNTER. St John's College, Cambeidge, October, 1865. CONTENTS. CHAPTER I. PAGE OF THE NATURE AND ORIGIN OF DIFFERENTIAL EQUATIONS 1 Definition, 2. Species, Order and Degree, 3. General Solution, Com- plete primitive, 6. Genesis of Differential equations, 8. How many of each order possible, — how many independent, 15. Cri- terion of derivation from a common primitive, 16, General form of equations thus related, 17. Geometrical illustrations, 18. Exercises, 20. CHAPTER II. ON DIFFERENTIAL EQUATIONS OF THE FIRST ORDER AND DEGREE BETWEEN TWO VARIABLES . . 23 General equation Mdx + Ndy = 0,23. Complete primitive f{x,y) = c,26. Geometrical illvistration, 28. Various integrable cases, 29. Ho- mogeneous equations, 34. Linear equations of first order, 38. Gather forms, 40. Solution by development, 41. Exercises, 44. CHAPTER III. EXACT DIFFERENTIAL EQUATIONS OF THE FIRST DEGREE 47 General criterion, 47. Mode of solution, 50. Practical simplifica- tions, 52. Exercises, 53. XU CONTENTS. CHAPTER lY. ON THE INTEGRATING FACTORS OF THE DIFFERENTIAL EQUATION Mdx + JSfdy =0, . . Integrating factors always exist, 56. General form of integrating factors, 57. Special determinations for homogeneous equations and for equations of tlie form F-^ {xy)ydx + F^{xy)xdy = 0, 69 — 62. Exercises, 67. CHAPTER Y. ON THE GENERAL DETERMINATION OF THE INTEGRATING FACTORS OF THE EQUATION Mdx + Ndy = . Partial differential equation for integrating factors, 69. Solution when the integrating factor is a function of x, 70 — of y, 71 — of xy, 72. When homogeneous, 74 — 76. More general application, 82. So- lution of P^dx + P^dy + Q, [xdy - ydx) = 0, 84. Jacobi's equation, 85. Euler's method, 86. Exercises, 88. CHAPTER YI. ON SOME REMARKABLE EQUATIONS OF THE FIRST ORDER AND DEGREE . . . .91 Kiccati's equation ^ + hu^ = cx"^, 91. x^-ay + ly"^ = cx^, 92. Solu- tion by continued fractions, 96, 97. Eiccati's equation made linear, 103. Euler's equation, 104. Theorem of development, 107. Exercises, 110. CHAPTER YII. ON DIFFERENTIAL EQUATIONS OF THE FIRST ORDER, BUT NOT OF THE FIRST DEGREE . . .113 Typical form, 113. Theory of its solution, 115. Kelation of complete primitive to particular integrals, &c., 117. Special methods, 121. One variable only involved, 122. x^{p) +y\{y(p) = x{p)i 124. Clairaut's equation, ib. Singular solutions, 125. Homogeneous equations, 128. Equations solvable by differentiation, 131. Trans- formations, 134. Exercises, 136. CONTENTS. Xlll CHAPTER yill. PAGE ON THE SINGULAR SOLUTIONS OF DIFFERENTIAL EQUATIONS OF THE FIRST ORDER . . . .139 Primary definition — ^positive and negative marks, 139, 140. Deriva- tion of the singular solution from the complete primitive, 141. General theorem, 148. Geometrical interpretation, 152. Deri- vation of the singular solution from the differential equation, 153. More general definition of singular solution, 163, Distinction of species, ib. General theorem, 164. Eeview of prior methods, 174. Properties of singular solutions, 177. Exercises, 182. CHAPTER IX. ON DIFFERENTIAL EQUATIONS OF AN ORDER HIGHER THAN THE FIRST . . . .187 Kelation to complete primitive, 187. Solution by development, 189. Linear equations, 192 — 199. General rule when the second member is 0, 200. Second member a function of x, — Variation of parameters, ib. Dependent class, 203. Properties of linear equations, 204. Analogy with Algebraic equations, 206. Exer- cises, 207. , "\ , CHAPTER X. « EQUATIONS OF AN ORDER HIGHER THAN THE FIRST, CONTINUED ^9 One variable wanting, 209. One differential coefficient present, 211. d-fh^y^^^- d-f\-S^y^^- Homogeneous eq«a- tions, 215. Exact equations, 222. Miscellaneous methods and examples, 226. Singular integrals, 229. Exercises, 234. CHAPTER XI. GEOMETRICAL APPLICATIONS . . . 238 Different problems, 239 — 244. Trajectories, 245. Curves of Pursuit, 251. Solution of a differential equation, 254. Involutes, 256. Inverse problem of caustics, 258 — 263. Direct problem, 259. Intrinsic equation of a curve, 263. Exercises, 269. XIV CONTENTS. CHAPTEE XII. PAGE ORDINARY DIFFERENTIAL EQUATIONS WITH MORE THAN TWO VARIABLES . . . .272 Isleajung of Pdx+Qdy+Rdz = 0, 272. Condition of derivation from a single primitive, 275. Solution, 276. General rule and ex- amples, 279. Homogeneous equations, 281. Integrating fac- tors, 282. Equations not derivable from a single primitive, 283. More than three variables, 286. Equations of an order higher than the first, 289. Exercises, 291. CHAPTER XIII. SIMULTANEOUS DIFFERENTIAL EQUATIONS . .292 Meaning of a determinate system, 292. General theory of simulta- neous equations of the first order and degree, 293 — 307. Systems of two equations, 294. Of more than two, 298. Linear equations with constant coefficients, 300. Equations of an order higher than the first, 307. Exercises, 316. CHAPTEH XIY. OF PARTIAL DIFFERENTIAL EQUATIONS . .319 Nature, 319. Primary modes of genesis, 321. Solution when all the differential coefficients have reference to only one of the inde- pendent variables, 322. Linear equations of first order, 324. Their genesis, 325. Their solution, 329. Non-linear equations of the first order, 335. Complete primitive, general primitive, and singular solution, 339. Sufficiency of a single complete pri- mitive, 345. Singular solutions, 346. Geometrical applica- tions, 347. Symmetrical and more general solution of equations of the first order, 350. Exercises, 358. CHAPTER XY. PARTIAL DIFFERENTIAL EQUATIONS OF THE SECOND ORDER 361 The equation Rr + Ss + Tt= V, 361. Condition of its admitting a first integral of the tovmu=f{v), 362. Deduction of such integral when possible, 364. Eelations of first integrals, 366. General rule, 369. Miscellaneous theorems. Poisson's method, 375. Duality, 376. Legendre's Transformation, 379. Exercises, 380. CONTENTS. XV CHAPTER XYI. PAGE SYMBOLICAL METHODS . . .381 Laws of direct expressions, 381 — 384. Inverse forms, 385. Linear equations with constant coefficients, 388. Forms purely sym- bolical, 398. Equations solvable by means of the properties of homogeneous functions, 403. The method generalized, 406. Exercises, 410. CHAPTER XVII. SYMBOLICAL METHODS, CONTINUED . .412 Symbolical form of differential equations with variable coefficients, 412. Finite solution, 415 — 436. Reduction of binomial equations, 418. Pfaff's equation, 430. Equations not binomial, 432. Solution by series, 437. Evaluation of series, 441. Generalization, 446. Theorem of development, 447. Laplace's reduction of partial differential equations, 450. Miscellaneous notices, 454. Exer- cises, 457. CHAPTER XYIII. SOLUTION OF LINEAR DIFFERENTIAL EQUATIONS BY DEFINITE INTEGRALS . . . .461 Laplace's method, 461. Partial differential equations, 475. Parseval's theorem, 477. Solution by Fourier's theorem, 478. Miscellaneous exercises, 481. The following 'portions of the work are recommended to beginners. Chap. I. Arts. 1—8, 11. Chap. ILArts. 1—11. Chap.IH. Chap. IV. Arts. 1—9. Chap. Y. Arts. 1—4. Chap. VI. Arts. 1—10. Chap. VII. Arts. 1—8. Chap. Vni. Arts. 1—7. Chap. IX. Arts. 1, 3—12. Chap. X. Arts. 1—3. Chap. XL Arts. 1—7. Chap. Xn. Arts. 1—8. Chap. XIII. Chap. XIV. Arts. 1—12. Chap. XV. Arts. 1—8. Chap. XVI. Arts. 1—8. Chap. XVII. Arts. 1—12. Chap. XVIII. Arts. 1—8. DIFFEKENTIAL EQUATIONS. CHAPTER I. OF THE NATURE AND ORIGIN OF DIFFERENTIAL EQUATIONS. 1. What is meant by a differential equation? To answer this question we must revert to the fundamental conceptions of the Differential Calculus. - The Differential Calculus contemplates quantity as subject to variation; and variation as capable of being measured. In comparing any tvfo variable quantities x and y connected by a known relation, e.g. the ordinate and abscissa of a given curve, it defines the rate of variation of the one, y^ as referred to that of the other, ic, by means of the fundamental con- ception of a limit; it expresses that ratio by a differential coefficient —■ ; and of that differential coefficient it shews how ax to determine the varying magnitude or value. Or, again, con- siderinsr ^ as a new variable, it seeks to determine the rate "=• dx ■ ' of its variation as referred to the same fixed standard, the variation of x, by means of a second differential coefiicient -^4 J and so on. But in all its applications, as well as in its theory and its processes, the primitive relation between the variables x and y is supposed to be known. In the Integral Calculus, on the other hand, it is the rela- tion among the primitive variables, x, y, &c. which is sought In that branch of the Integral Calculus with which the student B. D. E. 1 2 OF THE NATURE AND ORIGIN [CH. I. is supposed to be already familiar, the differential coefficient — being given in terms of the independent variable a?, it is proposed to determine the most general relation between y and X. Expressing the given relation in the form J=^(-) «' the relation sought is exhibited in the form y=\^{x)dx-\-c. In (1) we have a particular example of an equation in the expression of which a differential coefficient is involved. But instead of having as in that example ~ expressed in terms of X, ^Q might have that differential coefficient expressed in terms of y, or in terms of x and y. Or we might have an equation in which differential coefficients of a higher order, (3^11 d^ii -j^, -~, &c., were involved, with or without the primitive variables. All these including (1) are examples of differential equations. The essential character consists in the presence of differential coefficients. The equations ^ + 2^=0 (2) ^'^ + ^J + y = sina? (^)' are seen to be differential equations, the latter of which con- tains, while the former does not contain, the j)riniitive vari- ables. And thus we are led to the following definition. ' Def. a differential equation is an expressed relation in- volving differential coefficients, tuith or without the primitive variables from which those differential coefficients are derived. ART. 2.] OF DIFFERENTIAL EQUATIONS, 8 That which gives to the study of differential equations its peculiar value, is the circumstance that many of the most im- portant conceptions of Geometry and Mechanics can only be realized in thought by means of the fundamental conception of the limit. When such is the case, the only adequate ex- pression of those conceptions in language is through the me- dium of differential coefficients, — the only adequate expression of the truths and relations of which they are the subjects is in the form of differential equations. Sjpecies, Order and Degree. 2. The species of differential equations are determined either by the mode in which differential coefficients enter into their composition, or by the nature of the differential coeffi- cients themselves.- We may thus distinguish two great pri- mary classes of differential equations, viz. : 1st. Ordinary differential equations, or those in which all the differential coefficients involved have reference to a single independent variable. 2ndly. Partial differential equations, characterized by the presence oi partial differential coefficients, and therefore in- dicating the existence of two or more independent variables with respect to which those differential coefficients have been formed. Thus an equation such as (2) or (3), involving no other dv d 11 differential coefficients than -j- , -y^ > &c. is an ordinary dif- ferential equation, in which x is the independent, y the de- pendent variable. An equation involving ~ and -j- would, on the contrary, be a partial differential equation, having z for its dependent, a) and y for its independent variables. The equation a?y- + 2/-7-=^isa partial differential equation. The present chapter will be chiefly devoted to the con- sideration of that class of ordinary differential equations in 1—2 4 SPECIES, OKDER AND DEGKEE. [CH. I. which there exists a single independent variable x, a single dependent variable y^ and one or more of the differential coefficients of y taken with respect to x ; the presence of the last element only, viz. the differential coefficient, being essen- tial (Art. 1). The two following equations, in addition to those already given, will exemplify some of the chief varieties of the species under consideration : ^ ^ dx dy c (4), dx) ) /-N - —mx i'JJ. dx^ In (4) the independent variable x, the dependent variable y, and the differential coefficient -^ are all involved ; but, while in the previous examples J^ appears only in the first degree, in the present one it appears in the second degree and under a radical sign. In (5) we meet with the second differential coefficient ^4 in addition to the first differential coefficient v^ and the independent variable x. ax The typical or general form of a differential equation of the species just described is f(x v^^ ^ ^]=^0 (6), -^ V"' ^' dx' dx" ' dxV ^ ^' with the condition, already referred to, that one at least of the differential coefficients must explicitly present itself ^ All the above equations may at once be referred to the typical form by transposition of their second member. ART. 3.] SPECIES, ORDER AND DEGREE. 5 ^S. Differential Equations are ranked in oi^der and degree according to the following principles. 1st. The order of a differential equation is the same as the order of the highest differential coefficient which it con- tains. 2ndly. The degree of a differential equation is the same as the degree to which the differential coefficient which marks its order is raised, that coefficient being supposed to enter into the equation in a rational form. Thus the equation \dxj ax is of the first order and of the second degree. The equation d^u dy 7 2 is of the second order and of the first degree. The equation IV(^-49 (^)^ reduced to the rational form 2y-^4-^=°-" («)• is seen to be of the first order and second degree. The ground of the preference which is to be given to rational forms in the expression and in the classification of differential equations is, that a rational form is at the same time the most general form of which an equation is sus- ceptible. Thus (8) includes both the equations which would be formed by giving different signs to the radical in (7). The typical form of an ordinary differential equation of the first order is evidently A^'V't)-' (^)- 6 SPECIES, OEDEE AND DEGEEK fCH. I. ^ .... 4. "When a differential equation is capable of being ex- pressed in the form il + ^>i^ + ^i'- + ^"3/ = X (10). in wbicli tlie coefficients X^, X^, . . . X„ and the second member X are either constant quantities or functions of the indepen- dent variable x only, the equation is said to be linear. Equa- tions (1), (2) and (3) are thus seen to be linear, but (4) and (5) are not linear. If we refer (3), after dividing both members by i^, to the general form (10), we have n = % X.^^- , X=\,xJ^ . \Aj \Aj tAj "When the coefficients X^, Xg, &c. in the first member of a linear differential equation referred to the above general type are constant quantities, the equation is defined as a linear differential equation with constant coefficients. When those coefficients are not all constant it is defined as a linear dif- ferential equation with variable coefficients. The distinction is illustrated in the following examples : d^v _ cZV ^ dy ^ -y4 - 2 -r42 + 5 T^ - 8V = sm .T, ax ax ax ,^ „. d^y dy . the former of which is a linear differential equation with constant coefficients, while the latter would be described as a linear differential equation with variable coefficients. Meaning of the terms * general solution^ ' complete priinitive.' 5. In all differential equations there is, as has been seen, an implied reference to some relation among variable quantities dependent and independent; such reference being established through the medium of differential coefficients. Now the chief object of the study of differential equations is to enable us to ART. 5.] GENERAL SOLUTION. 7 determine whenever it is possible, and in the most general manner which is possible, such implied relation among the primitive variables. That relation, when discovered, is, by the adoption of a term primarily applicable to the mode or process of its discovery, called the solution of the equation. Thus if the given equation be cc -^ -\- y = cos^ -(ll)? the following process of solution may be adopted. Multiply- ing by dsc, we have ocdy + ydx — cos xdsc^ and integrating, since each member is an exact differential, xy — sina;+ c (^2). The result is termed the solution, or, still more definitely, the general solution of the equation. It involves an arbitrary constant, c, by giving particular values to which a series of particular solutions is obtained. The equations xy = sin a;, xy — sin ic + 1, are particular solutions of the given differential equation. The term solution is still employed, even when the inte- gration necessary in order to obtain in a finite and explicit form the relation between the variables cannot be effected. Thus if we had the differential equation, j£-y-xe'^i) (IS),. we should thence derive in succession xdy — ydx _ d'dx ^ e'^dx a? X 1 = ['. X J X + c (U), 8 GENESIS OF DIFFERENTIAL EQUATIONS. [CH. I. and the last result is called the solution of the given equation,- although it involves an integration which cannot be performed in finite terms. The relation among the variables which constitutes the general solution of a differential equation, as above described, is also termed its complete primitive. The relation (14) in- volving the arbitrary constant c is virtually the complete primitive of the differential equation (13). It will be observed that the terms 'general solution' and * complete primitive/ though applied to a common object, have relation to distinct processes and to a distinct order of thought. In the strict application of the former term we contemplate the differential equation as prior in the order of thought, and the explicit relation among the variables as thence deduced by a process of solution; while in the strict use of the latter term the order both of thought and of process is reversed. Genesis of Differential Equations, 6. The theory of the genesis of differential equations from their primitives is to a certain extent explained in treatises on the Differential Calculus, but there are some points of great importance relating to the connexion of differential equations thus derived, not only with their primitive, but with each other, which need a distinct elucidation. Suppose that the complete primitive expresses a relation between x, y and an arbitrary constant c. Differentiating on the supposition that x is the independent variable, we obtain a new equation which must involve ~- , and which may involve any or all of the quantities x, y and c. If it do not involve c, it will constitute the differential equation of the first order corresponding to the given primitive. If it involve c, then (the elimination of c between it and the primitive will lead to jthe differential equation in question. Thus if the complete primitive be y=^cx ,,,... (1), ART. 6.] GENESIS OF DIFFERENTIAL EQUATIONS. 9 we have on differentiation, I- •••(^)' and, eliminating the constant c, ^=^i-" ^^)' the differential equation of the first order of which (1) is the complete primitive. That primitive might have been so prepared as to lead to the same final equation by mere differentiation. Thus, re- ducing the primitive to the form we have on differentiating and clearing the result of fractions, dx ^ which agrees with (3). And generally, if a primitive involving an arbitrary constant c be reduced to the form (a, 6), is itself an arbitrary constant, and may be represented by c. Hence any equation of the form *W'^'2''i)' ^("'2/,i)} = c (7) would, equally with (5) and (6), constitute a first integral of the supposed equation of the second order. It is evident that (7) is the general type of all such first integrals. Thus the type of the first integrals of (3) would be \x dx' ^ ^ dx) But any two first integrals included under this type and in- dependent of each other would lead us, as is obvious, to the same final integral (4), either under its actual or under an equivalent form. While therefore, viewed as an independent system, the first integrals of a differential equation of the second order are but B. D. E. ■ 2 18 GEOMETEICAL ILLUSTRATIONS. [CH. I. two, it is formally more correct to regard them as infinite in number, but as so related that any two of them which are independent contain by implication all the rest. Such considerations are easily extended to differential equations of the higher orders. Geometrical illustrations, ^ 11. Geometry, by its peculiar conceptions of direction, tangency, and curvature, all developed out of the primary conception of the limit. Art. 1, throws much light on the nature of differential equations. As the simplest illustration let the equation of a straight line y = ax-\-h (1) be taken as the complete primitive, a and h being arbitrary constants. Differentiating, we have d-^^" (2). Eliminating a, we find . , y-4.=' (•^)' and again differentiating . s-» (* Of these equations, (4), which is free from arbitrary con- stants, is the general differential equation of the second order of a straight line; and (2) and (3), each of which contains one of the original arbitrary constants, are the two differential equations of the first order. Moreover, each of these dif- ferential equations expresses some general property of the straight line — (2), that its inclination to the axis is uniform ; (3), that any intercept, parallel to the axis of y^ between the ART. 11.] GEOMETRICAL ILLUSTRATIONS. 19 straight line and a parallel to it through the origin will be of constant length; (4), that a straight line is nowhere either convex or concave ; — and this property, which does not in- volve, in the same definite manner as the others do, the con- siderations of distance and of angular magnitude, is evidently the most absolute of the three. The equation of the circle is {x-ay+{y-hf = r' (5), and if we regard a and h as arbitrary constants the corre- spondiug differential equation of the second order will be \dx . dx'^ = »• (6), expressing the property that the radius of curvature is in- variable and equal to r. If we proceed to another differentiation, we find which is the general differential equation of a circle free from arbitrary constants. And the geometrical property which this equation also expresses is the invariability of the radius of curvature, but the expression is of a more absolute character than that of the previous equation (6). For in that equation we may attribute to r a definite value, and then it ceases to be the differential equation of all circles, and pertains to that particular circle only whose radius is r. The equation (7) admits of no such limitation. Monge has deduced the general differential equation of lines of the second order expressed by the algebraic equation ax^ + hxij -h cif 4- ex -{-fu — 1. O 9 20 EXERCISES. [CH. I. It is \dx^J dx^ daf dx^ dx^ \ dxV ' But here our powers of geometrical interpretation fail, and results such as this can scarcely be otherwise useful than as a registry of integrable forms. From the above examples it will be evident that the higher the order of the differential equation obtained by eli- mination of the determining constants from the equation of a curve, the higher and more absolute is the property w^hich that differential equation expresses. We reserve to a future Chapter the consideration of the genesis of partial differential equations as well as of ordinary differential equations involving more than two variables. EXERCISES. 1. Distinguish the following differential equations accord- ing to species, order, and degree, and take account of any peculiarities dependent upon their coefficients. (1) J^-x^y = ax\ (2) -nz + --T-- ^y = 0. ^ ^ dx' xdx ^ dz dz f^ (4) X-, V — = — . ^ dx ^ dy y ,^. - dhc d^u d'u „ ^•■'^ d^' + df + rf? = ^- CH. I.] EXERCISES. 21 2. Explain the term 'complete primitive,' and form the differential equations of the first order of which the following are the complete primitives, c being regarded as the arbitrary constant, viz. : (1) y=cx + ^/{\ + c'). (2) y = (« + c)e» , . (3) y = ce"**" ^+ tan"^ x — 1. (4) • 2/ = (ex ■{• log X + 1)~\ (5) y'-^cx-c'^O. (6) 2/=c^ + (/)(c). , OyowV\:c)o-tCN^ 3. Form the differential equations of the second order of which the following are the complete primitives, c and c being regarded as arbitrary constants. (1) y — G cos mx + c sin mx. (2) y = c co^ (mx-\-c'). c + ex (3) 2/=ajlog a? sin mx (4) 2/ = c sin nx + ,g' cos 7i^ + 2m 4. State the criterion by which it may be determined whether differential equations are derived from a common primitive. 5. Shew that the differential equations are not derived from a common primitive involving a and h as arbitrary constants. 6. Shew that each of the following pairs of equations, in lich p stands for -j- , is deri and determine the primitive : Qll , ... which J) stands for -j- , is derived from a common primitive 22 EXERCISES. [CH. I. (2) y — xp = a {y^ +^), and y — 0Dp = h(l+ x^p). 7. How many first, second, third, &c. integrals belong to the general differential equation of lines of the second order given in Art. 11, and how many of each order are inde- pendent? 8. From the equation (y — 6)^= 4m (x — a) assumed as the primitive, deduce 1st the differential equations of the first order involving a and h as their respective arbitrary constants ; 2dly the general functional expression for all differential equa- tions of the first order derivable from the same primitive. 9. Of what primitive involving two arbitrary constants would the functional equation ^(y — 2px, p^x) = c represent all possible differential equations of the first order ? 10. How many independent differential equations of all orders are derivable from a given primitive involving x, y, and n arbitrary constants? ( 23 ) CHAPTER II. ON DIFFERENTIAL EQUATIONS OF THE FIRST ORDER AND DEGREE BETWEEN TWO VARIABLES. 1. The differential equations of wliicli we shall treat in this Chapter may be represented under the general form OjCG M and N being functions of the variables x and y. In this mode of representation x is regarded as the inde- pendent variable and y as the dependent variable. "We may, however, regard y as the independent and x as the dependent variable, on which supposition the form of the typical equation will be dy For as any primitive equation between x and y enables us theoretically to determine either y as a function of x, or x as a function of y, it is indifferent which of the two variables we suppose independent. It is usual to treat this equation under the form Mdx-\-Ndy = <^, not however from any preference for the theory of infinitesi- mals, but for the sake of symmetry. The order of this Chapter will be the following. As the solution of the equation, if such exist, must be in the form of a relation connecting x and y, I shall first establish a prelimi- nary proposition expressing the condition of mutual depend- 24 DIFFEEENTIAL EQUATIONS OF THE [CH. II. jj- eace of functions of two variables ; I shall then inquire what kind of relation between x and y is necessarily implied by the existence of a diflferential equation of the form 2ZZ I shall discuss certain cases in which the equation admits readily of finite solution; and I shall lastly deduce its general JT solution in a series. Peop. 1. Let V and v he exj^Ucit functions of the two vari- ables X and y. Then, if V he expressible as a function of v, the condition dVdv dVdv _ , . dx dy dy dx " ^ ^ will he identically satisfied. Conversely, if this condition he identically satisfied^ V will he expressible as a function ofv, 1st. For suppose V= (j> (v). Then dV _ d(j)(v) dv dx dv dx' dV _ d(f> (v) dv dy^ dv dy * dv dv Multiplying the first equation by -7- , the second by -,- , and subtracting, we have dVdv dV dv _ dx dy dy dx And this is satisfied identically; since by the process of elimination the second member vanishes independently both of the form of ^ as a function of x and y, and of the form of F as a function of v, 2ndly. Also if the above condition be satisfied identically, Fwill be expressible as a function of v. For whatever func- tions F and tj may be of x and y, it will be possible by elimi- ART. 1.] FIRST ORDER AND DEGREE. 25 nating one of the variables x and y to express F as a function of the other variable and of v. Suppose for instance the expression for F thus obtained to be .^., V= (x, v) ^ ^ dx identically. Therefore cj) (x, v), which represents V, does not contain x in its expression ; and F reduces simply to a func- tion of V. We have supposed each of the functions F and v to con- tain both the variables x and y. But, whether this be or be not the case, the identical satisfying of (1) is the necessary and sufficient condition of the functional dependence of F and V. For suppose either F or v, and for distinction we shall choose V, to be a function of one of the variables only, as x, and F to be a function of v. Then is F also a function of x, and as -r- and -y- vanish identically the condition (1) is satis- dy ay *^ fied. f 26 DIFFERENTIAL EQUATIONS OF THE [CH. II. Conversely, supposing i? to be a function of w only, and (1) to be identically satisfied, that equation reduces to wbence V is expressible as a function of v. 2. The equation M-\- N -/- = always involves the existence of a primitive relation between x and y of the form fix, y) = c, in which c is an arbitrary constant. Let us first consider what is the immediate signification of the equation M+N^^^O ..(1). We know that if Aa? represent any finite increment of cc, and A?/ the corresponding finite increment of y, -^ will represent A'?/ the limit to which the ratio -r^ approaches as Aaj approaches toO. Let us then first examine the interpretation of the equation ^+i^^ = ...(2). We have -7r- — —^r' The second member of this equation being a function of x and ?/, since M and iV are functions of those variables, we may write || = 'A(^.2/) (3), the form of ^ (tc, 3/) being known when M and H are given. Now if we assign to x any series of values, it is possible to assign a corresponding series of values of y, any one of which being fixed arbitrarily all the others will be determined by (3). AET. 2.] FIRST ORDER AND DEGREE, 27 Thus let Xq, x^, cCg... be the series of arbitrary values of a?, and y^ an arbitrary value of y corresponding to x^ as the value of X, then, representing by ^x^ the increment of x^, i.e. the value which being added to x^ converts it into x^, we have by (SJ "^^^-'l %o == 4> ip^s^^ i/o) ^^0 ; JjC-^ therefore y, + Ay,^y^ + ^ {x„ y,) Ax,. But as Ay, represents the increment of y, corresponding to AXq as the increment of x^^ it is evident that y, + Ay, will be the value of y corresponding to x, + Ax, as the value of x, Representing then this value of y by y^ we shall have 2/1 = 2/0 + ^(^0.2/0)^^0 ==yo + ^K. yo)K-^o) (4). In like manner we shall find 2/2=^1 + ^(^1' 3^1) K-^i) (5), but, 2/1 being already determined by (4), y^ is determined, and continuing the operation, a series of values of y will be deter- mined, only one of which is arbitrary, while all the others are assigned in terms of that arbitrary value and of the known values of x. If, for example, we have the particular equation Ay = (x + y) Axj and assign to x the series of values 0, 1, 2, 3, 4, &c., and at the same time assume that when x is equal to 0, y is equal to 1, we shall have the two following corresponding series of values, viz. Xq = 0, ^j = 1, x^ =2, ^3 = 3, x^ — 4, &c. 2/0 = 1, 2/1 = 2, 2/2=5' 2/3=12, 2/4=27, &c. By assigning a different value to y,, or by assuming arbi- trarily the value of some other term of the series y,, y^, y^, &c. we should find another set of values of those quantities cor- responding to the given values of x. But, in every such set, 28 DIFFERENTIAL EQUATIONS OF THE [CH. II. the values of all the terms but one will be determined by a law. Now if the intervals between the successive values of x are diminished, while the number is proportionately increased, each of the corresponding sets of values of x and y will more and more approach to the state of continuous magnitude. And, in the limit, to every conceivable value of x will corre- spond a value oiy, determined in subjection to a continuous law — to a law however which permits us to assign one of the values of y arbitrarily. The analytical expression of that law will be the solution of the di:^erential equation given. 3. To illustrate the same doctrine geometrically, if x and y represent rectangular co-ordinates, any system such as the above would represent a series of points of which the abscissas having been assumed arbitrarity, the corresponding values of 2/, except one, are determined by a continuous law. In the limit, that series of points would approximate to a curve the species of which as dependent upon the form of its equation would be determined by a law, but an element of which, re- presented by a constant in that equation, would be left arbi- trary, so as to permit us to draw the curve through a given point. The form of the analytical solution thus indicated is f{^>y)=o (6). The genesis of differential equations of the first order and degree from equations of this description has already been explained in Chap. I. Art. 6. It is evident that, as c is arbi:: trary, such a value may be assigned to it as to make a given value of y correspond to a given value of cc. If those corre- sponding values are x^, y^, we have only to assume /(^o.2/o) = c (7)y whence c is determined. But c beiug once determined, all the values of y depend upon those of x, in obedience to the law expressed by (6). ART. 4.] FIRST ORDER AND DEGREE. 29 Lastly it may be sbewn that two distinct complete primi- tives of Mdx + JSI'd?/ = cannot exist. For suppose that there are two such primitives then by differentiating each du dudy _ ^ dv dv dy _^^ dx dydx ' dx dydx ' whence, eliminatins^ -^ , ' ^ dx* du dv du dv _ dx dy dy dx ' which shews, by Prop. I, that v is a function of u. The second equation is then equivalent to and this is resolvable by solution into equations of the form each of which is therefore only a repetition of the first sup- posed complete primitive. Certain cases in which the equation Mdx+Ndy — admits of finite solution. 4. The equation Mdx + Ndy = can always he solved when the variables in M and N admit of being separated ; i. e. when the equation can be reduced to the form Xdx + Ydy = (8), in which X is a function of x alone, and Y a function of y alone. To solve the equation in its reduced form (8), it is only necessary to integrate the two terms separately, and to equate the result to an arbitrary constant. Thus the solution will be jxdx+JYdy = c (9). 30 DIFFEEENTIAL EQUATIONS OF THE [CH. II. On differentiating this result the arbitrary constant c dis- appears, and (8) is reproduced. Thus the solution of the equation xdx + ydy = 2 1 2 will be — 2 ^ ^> or, since c is arbitrary, The solution of the equation dx dif f. l + aj 1 +y will in like manner be log (1 + ^) + log (1 -{- ?/) = c ; a result which may be simplified in the following manner. We have log (1 + a?) (1 + 2/) = c ; therefore (1 -\-x) (1 + y) = e'. But a function of an arbitrary constant is itself an arbitrary constant. Hence we may write as the solution {l+x){l + y)=G. Indeed it frequently happens that solutions which present themselves in a transcendental form admit of being reduced to an algebraic form. Thus also the solution of the equation ^•^ +— ^ = (10) V(i-^^) VCi-2/'J being -1 sm X + sm '^ y— c, we shall have, on taking the sine of both members of the equation and replacing sin c by (7, ^V(i-2/')+W(i-^')=^ (11). which is algebraic. ART. 5.] FIEST ORDER AND DEGREE. 81 5. Different modes of integration will also give rise to solutions which at first sight appear to be discordant. The discordance however will be only apparent. Thus if we ex- press the equation last solved in the form and integi'ate by means of the formula — dx h = cos o) + const., we shall have cos~^ X + cos~^ y = ^1 and, taking the cosine of both members, ^ xy-^{{\-x'){l-y')] = coBG, (12). The last result may however be reduced to the form xsl(l-f)^ysl{l-x') = B\iiC^ (18), which, as sin G^ is arbitrary, agrees with the previous re- sult, (11). The constants C and C^ are seen to be connected by a relation (7= sin (7^, which is independent of the variables x and y. And in general the test of the accordance of two solutions of a differential equation, each involving an arbitrary constant, is, that on eliminating one of the variables, the other variable tuill disappear also, and a relation between the arbitrary con- stants alone residt. Or expressing the solutions in the form we may directly apply the test of equivalence dVdv dVdv __ ^ dx dy dy dx ' resulting from the proposition in Art. 1. 32 DIFFERENTIAL EQUATIONS OF THE [CH. II. 6. It sometimes happens that the variables may be sepa- rated by multiplying or dividing the equation by a factor. Thus the equation xdx ydif 1+2/ 1 -\-x becomes on multiplying by (1 + a?) (1 + y), X {1 + cc) doo — y (1 +7/) dy = 0, in which the variables are separated. Integration then gives 2 3 2 3 ' X X y y _ The most general form of equations in which the variables can be separated by the process above mentioned is X^Y^dx-\-XJ^dy=:0 (14), in which X^ and X^ are functions of x only, and Y^ and Y^ functions of y only. On dividing the above equation by YjXg, or, which amounts to the same thing, multiplying it by 1 the factor ^ y , we have ■^ 12 ^dx-\-^dij=^0 (15), in which the variables are separated. Ex. The equation x \/[l + ?/^) dx-^y \f(l + x^) dy — is thus reduced to xdx ydy __ „ and has for its complete integral 7. Sometimes too the variables in the equation Mdx^Xdy = admit of being separated after a preliminary transformation.- I ABT, 7.] FIRST OEDER AND DEGREE. S8 Ex. 1. If in the equation {x — y^) dx + 2xydy = 0, we assume y = ^/{xz)f we find a^ --r > '^ , zdx + xdz X y jLf - v.^ / a^^^ Substituting these expressions for y and dy in the given , equation, we have . >^^ '^ ' X -x Therefore integrating and replacing z by its value — , X ° X Ex. 2. Q/-^) (1 + x')^ dy-n(l -{-y'fdx = 0. Assume x — tan 6, y — tan ^. "We find (tan -nde = 0. Now let — nd-^y therefore dA = r — r ; ^ ?z — sm Y whence {v)+V^jr {v) ^ ^ It is obvious from the symmetry of the relation between x X and y that we might equally employ the transformation - = v and regard v and y as the new variables. What is essential in the method is the substitution, in place of the original vari- ables X and y, of a new system of variables, consisting of one variable of the old system, and of the ratio which is borne to it by the other variable of that system. Ex. It is required to integrate the equation . {x - V(^y) -y] dx-{- \/{xy) dy = 0, 3—2 36 HOMOGENEOUS EQUATIONS. [CH. n. by the direct application of (18). Here, n = 1, M~x — fjipcy) — 2/ = iK {1 — hjiy) — v}y JV= /^(ocy) = X VW« Thus we have yjr (v) = v^,^ and (18) gives %^+l 1 ^ = (^' •^ 1 — -y^ — -y + ^2 To effect the integration in the second term, let v=^f. 2fdt Then/ f'^ . = f t-f + f 1 1-t 1 f flog (l-0 + i% (1 + + log(i-0+ilog(l-O. Hence finally, replacing ^ by ^ , x^ — y'^ 9. The equation (ax-\-ly^-c)dx^r {ax-\-h'y-\- c) dy = fd (19) may be rendered homogeneous, either first by assuming x^x -(X, y-y - ^, and properly determining a and /5 ; or secondly by assuming ax+hy -\-c =x\ ax + h'y + c' = y\ AET. 9.] HOMOGENEOUS EQUATIONS. 37 The first transformation gives {ax + ly- aa-hl3 + c) dx + {ax + Vy - a'a - U^ + c) dy = 0, whence if a and ^ be determined by the conditions aa + 6/3 = c, a a + y^ — dy we shall have the homogeneous equation {ax + hy) dx + (aV + h'y) dy = 0. - Making then y = vx we find -^ + ^ + (6+a')^ + 6V~" ^ ''^ which is directly integrable. The second transformation gives adx + hdy — dx ^ adx + h'dy — dy\ whence determining dx and dy, the proposed equation _^ assumes the homogeneous form -i^' s^ w ^ {b'x — a'y) dx — {hx — ay) dy = 0. e-^-^ Both these transformations fail if ah' — ah — 0. But in this case, since h' = — ^ , the proposed equation may be expressed "^ ^ a T-'. in the form , -^ {ax + hy + c) dx + — (ax -{■ hy + —r] dy = 0, ^ -^ a \ a J and the variables will be separated if we assume ax-\-hy =■ z, and then adopt either z and a; or ^ and y as the new variables. These transformations are linear, and by one of the two the proposed equation is usually solved. [For another method seethe Supplementary Volume^ Chap- ^ ter XIX, Arts. 1 and 2:] 38 LINEAR DIFFERENTIAL EQUATIONS [CH. II. 10. The linear differential equation of the first order and degree 'S>+p^=Q- (21)' P and Q being functions of x, admits of being solved. When Q—0 the solution is obtained by separating the variables ; and -when Q is not equal to 0, a solution may be founded upon that of the previous and simpler case. It must be observed that the linear equation (21), when reduced to the form falls under the general type, Mdx + Nd^/ = 0. 1st, When Q = 0, we have dy dx + p2/ = 0. Dividing by y, in order to separate the variables, dy ^^-Pdx. Therefore, log y = — I Pdx + c, which gives y —. ^ -fPdX+C = (76--/'^'^^ (22), C being an arbitrary constant substituted for e". It has been Sflready observed that a function of an arbitrary constant is itself an arbitrary constant ; see Art. 4. 2ndly, To solve the linear equation (21) when Q is not equal to 0, let us assign to the solution the general form (22) above obtained, but suppose G to be no longer a constant but a new variable quantity — an unknown function of x, which must be ART. 10.] OF THE FIRST ORDER AND DEGREE. 89 determined in accordance with the new conditions to which the solution must be subject. Substituting then the above expression for y in (21), and observing that, since is now variable, we have dx ax dx there results dx fin Hence -f-^ef^^*Q, dx Therefore = [ e^^^^ Qdx'^ c. c being an arbitrary constant. Substituting this generalized value of G in (22), we have finally y^e-f^^-[\^f^''^qdx\c\ (23), the solution required. It will be observed that if Q = 0, the above solution is reduced to the form (22) before obtained. The method of generalizing a solution above exemplified is called the method of the variation of 'parameters, the term parameter, by an extension of its use in the conic sections, being applied to denote the arbitrary constants of the solution of a differential equation. It is only, however, in certain cases that this method is successful. It is always legitimate to endeavour to adapt a solution to wider conditions by a transformation, which, like the above, only introduces a new variable instead of an old one, or a new and adequate system of variables in the room of a former system. But it is not always that the equations thus obtained are, as in the above example, easier of solution than those of which they take the place. 40 LINEAR DIFFERENTIAL EQUATIONS [CH. II. Ex. 1. Given ^ %= = (a? + 1)^ Here P=^. ^ = (^ + 1^. <^'^ - ^ - Hence \Fdx = - 2 log (aj + 1), e^-^^ = (ic + 1)"^ [e/^'^* Qdx =j(x +l)dx= ^-—^ + c. Therefore y={x + ly jl^l^T + 1 . Ex. 2. Given ^ ^ = €=^ (^ + l)'*. do) x + 1 Here we find JFdx = — ti log (cc + 1), t6f^''^Qdx=je'dx=e\ Therefore y={x+ 1)" (e'' + c). 11. Equations of the form dy dx + Py^Qf, P and Q being functions of x, are reducible to a linear form. For, dividing by y'^y we have Now let 2/*^ = «, then ^ ^ '^ dx dx * , ^dy 1 dz whence V -r- — =i f ^ ^ dx 1—ndx so that the equation becomes ART. 12.] OF THE FIRST ORDER AND DEGREE. 41 1—ndx dx ov^ + {l-n)Pz=(l-n)Q, whicli is linear. ^^ ^'^^^5^ + ^Ti" 2 • Here, dividing by y\ we have ^ c^^"*"a; + l 2 ' and, assuming y~^ = z, ^5i"*"^Tl 2~' or ^_2-^ = (aj+l)^ The solution of this, equation, which is identical in form with that of Ex. 1, is whence g/ = J ^^ ^ j^c{w-\-Vf\ . General sohction hy development. 12. In the earlier portion of this Chapter it was esta- blished, by considerations founded upon the nature and inter- pretation of the equation Mdx + Ndy = 0, that it implied the existence of a primitive equation between aj, y, and an arbitrary constant. The examples of finite solu- tion which have been given above, illustrate this truth. But 42 GENEEAL SOLUTION BY DEVELOPMENT. [CH. IL a further and more complete illustration is afforded by tlie presence of an arbitrary constant in the general integral of the equation, as developed in the form of a series by Taylor's theorem. This mode of solution we now proceed to exhibit. From the given equation we have the second member of which, being a function of a? and^, may be represented by/j (x, y). Thus we may write J=/.(«'.^) (24). t And differentiating this equation . <^V^^/i(^. y) I ^i(^^ y) dy da? dx dy dx ='^^^/^(-^)' - the second member of which, being a function of x and y, may be represented by f^ {x, y). Thus we have, as a conse- quence of (24), g=/.(^.2/) • (25). Repeating on this equation the above process of differentia- tion and substitution, we have wherein S=/3(^.2/) •••••(26), And, continuing thus to repeat the same operation, we obtain ART. 12.] .GENERAL SOLUTION BY DEVELOPMENT. 43 a series of equations determining the successive differential coefficients of y, in the form g=/„Ky) (27), . thfe dependence oi,f^ {x, y) upon f^_^ {x, y), and hence ulti- mately upon/j {x, y), being determined by the general equa- tion ' /»(-,2/) = ^#^^^^V.(-.2/) (28). . Hence M ajid N being given, the expressions for dy d^y dx' dx^' '" are implicitly given also. Now -T^ , -~, , &c. determine the coefficients of the several ■ dx dx^ terms after the first in the development of y in ascending j)owers of x, by Taylor's theorem, or more generally in as- cending powers of x — Xq, where x^ is a particular value of x. Leaving that first term arbitrary, the development is thus seen to be possible, and the result, while constituting the general integral of the given differential equation, shews that that integral involves an arbitrary constant. Actually to obtain the development, let ^(«) represent the general value of y, and let y^ be the particular value of ?/ corresponding to some 'particular and definite value, w^, of the variable a?. Then, writing ^ (cc) in the form (l>(x^ + x-xj, we have, by Taylor's theorem, y = 4, (x,) + ^•(x:)(x-x,) + i>"(x,) ^^^' + &c. ... (29). But ^ (Xq) is what y becomes when x = x^. Hence (a? J = y^. Again, (pXx^ is what ^^ , i.e. -^, becomes when x—x^. Hence ^'(^o) —fi(pPoy]/o) ^J (^4). In like manner ^"fe) ^^ 44 GENERAL SOLUTION BY DEVELOPMENT. [CH. IL what -y^ becomes when x — x^^ and is therefore equal to f^ (^o> y^' Determining thus the successive coefficients of (29), we have finally y^y^ +/i K. y^ (^ - «^o) +/2 fe> y^ x^i + &c. . . . (so), which is the general integral. If we assume x^ = 0, and represent the corresponding value of 2/ by c, we have 2^ = c+/,(0,c)aj+/,(0,c)j^ + &a (31). Should however any of the coefficients in this development become infinite we must revert to the previous form, and give to x^ such a value as will render the coefficients finite, and therefore justify the apphcation of Taylor's theorem. Virtually the integral (30) involves like (31) only one arbi- trary constant. For in applying it we are supposed to give to x^ a definite value, and this being done the corresponding arbitrary value of y^ constitutes the single arbitrary constant of the solution. [See the Supplementary Yolumet Chapter xix, Arts. 4 and 0.] EXERCISES. 1. Integrate the differential equations: . (1) (1 + x) ydx + (1 — ?/) xdy = 0. • (2) [y^^xf) dx + (oj' - yx^) dy=^0. (3) xy{l-\'X^)dy-'{l + f)dx^O. (4) [l^f)dx--[y-^^(l-^f)\{l + x'fdy = 0. (5) sin aj cos 2/cZa; — cos aj sin 2/c?2/ = 0. (6) sec^ictan^c?^ + sec^^tanaj% = 0. €H. II.] EXEECISES. 45 2. Different processes of solution present the primitive of a differential equation Under the following different forms, viz. tan~^ (x-{-y) + tan~* (^ "" ^) = c, Are these results accordant 1 3. Integrate the homogeneous equations ; (1) (y — x) dy + ydx = 0. (2) {2 V(%) ~ ^} % +2/^^ == ^• (3) oody — ydx — ^J{x^ + 3/^) cZ^ = 0, (4) ix — y co^-\ dx ■\- X ao^- dy — 0. \ XJ X (5) (8?/ + 10^) c?^ + i^y + 7^) c??/ =^-0i 4. Integrate the equations : (1) i^x-^y-\'V)dx^{^y-x-\)dy=^^. (2) (3y-Ya; + 7)^^+(7?/-3^ + 3)c?2/ = 0; the former as an exact differential equation, the latter by re- duction to a homogeneous form. 5. Explain what is meant by variation of parameters, and, du having integrated the equation x-J^ — ay—0, deduce by that dii method the solution of the equation; cc -^ — a?/ = cc + 1. 6. Integrate, by the direct application of (23), the linear equations, /-i\ dy ^ X 1 dx 1 + x' ^ 2^ (1 + x") ' (2) X (1 - x"") ^1 + (2^^ - 1)2/ = (ix\ , . dji y _ a? + VCl - o?) ^'^ dx^ i^^^-f- (1-c.y ' 46 EXERCISES. [CH. II. 7. Shew that the solution of the general linear equation -^ + Py = Q may be expressed in the form ((7+/e 8. Shew that, <^ (x) being any function of x, the solution of the linear equation ^-ycl>\x)==^{x)^\x), will he y = ce^^"^^ -(j){x)—l, 9. Shew that if in the linear equation ~- + Py = Q we represent ~ by p, and then, differentiating and eliminating y, form a differential equation between p and a;, that equation will also be linear. 10. Integrate the differential equations ; dz dz (2) S2'^-az'==x+l. dx (3) .^ + 2xz = "LaxW dz dx (4) -T- -^ z cos a; = s" sin 2ir. • (5) ^^ + 2/ = 2/' log a?. ( 47 ) CHAPTER III. EXACT DIFFERENTIAL EQUATIONS OF THE FIRST DEGREE. 1. As tlie cases considered in the previous Chapter under which the equation Mdx + Ndy = is integrable by the sepa- ration of the variables, are but a small number of the cases in Avhich a solution expressible in finite terms exists, Analysts have engaged in a more fundamental inquiry of which the following are the objects, viz. 1st, To ascertain under what conditions the equation Mdx + Nd7/=0 is derived by immediate differentiation from a primitive of the form / {x, y) = c, and how, when those conditions are satisfied, the primitive may be found. 2ndly, To ascertain whether, when those conditions are not satisfied, it is possible to discover a factor by which the equa- tion Mdx -h Ndy = being multiplied, its first member will become an exact differential. . These inquiries will form the subject of this and the follow- ing Chapter. . Prop. i. The one necessary and sufficient condition under which the first member of the equation Mdx + Ndy = is an exact differential is dM_dI^ , . dy dx Let it be considered in the first place what is meant by the supposition that Mdx + Ndy is an exact differential. It is that M and A^ are partial differential coefficients with respect 48 EXACT DIFFEEENTIAL EQUATIONS [CH. III. to X and t/, — thai there exists some function F, such that J=^- (2)' dV ^=N (3). dy Any relation between M and N" which we can derive inde- pendently of the form of V from the above equations will be a necessary/ condition oiMdx -^-Ndy being an exact differential, And conversely, any relation between if and iV which suffices to enable us to discover a function V actually satisfying the above equations (2), (3), will be a sufficient condition of Mdx + Ndy being an exact differential. And if the same condition should present itself in both cases, it will be both necessary and sufficient Differentiating (2) with respect to y^ and (3) with respect to Xy we have dr^^dM dW^dN dydx ~ dy ' dxdy dx '"* ^ ^' But the first members of these equations being, by a known theorem of the Differential Calculus, equal, we have dM_dF dy dx ^ '' This, therefore, is a necessary condition of Mdx + Ndy being an exact differential. It is also, as will next be shewn, a sufficient condition. In the first place the function F, if such exist, must satisfy the equation (2). Integrating this equation relatively to x alone (since the dV . differentiation in -t— is relative to x alone), we have V^lMdx^G ^hldx^ C (6), ART. ij OF, THE FIRST DEGREE. 49 G being a quantity which is constant relatively to x, so that dC -7- = 0. Hence, though G does not vary with x, it may vary CLX with y, and there is nothing to limit the manner of its varia- tion. It is therefore an arbitrary function of y, and we may write V=[Mda) + (j>{y) (7). This is the most general form of F as a function of x and y, which satisfies the equation (2). In the second place Fmust satisfy the equation (3). Sub- stituting in that equation the value of V given in (7), we have dfMdx ^ dcj,(y) ^^^ dy dy Therefore d^y)^^_dSMdx^ dy dy Whence 4,{y)=^[N-^^)dy + G (8), G being simply an arbitrary constant, since, as the constant of integration with respect to y it cannot contain y, and as part of the expression for ^ {y) it cannot contain x. Now the integration in the second member is theoretically possible (though its expression in finite terms may not be dfMdx possible) if the coefficient of dy, viz. I^ ~ — , is a function of y only, i.e. if its differential coefficient with respect to x is 0. Expressing this condition, we have dJSr d dfMdx _ . dx dx dy ^ ^ d dJMdx _ d djMdx dx dy dy dx . _dM ~ dy* B. D. E. 4 gives 50 EXACT DIFFERENTIAL EQUATIONS [CH. III. Thus the condition (9) becomes ^-<^=0 (10). ax ay This then is a sufficient, as it has before been shewn to be a necessary condition of Mdx + Ndy being an exact diffe- rential. The substitution in (7) of the value of ^ {y) found in (8) V=JMdx+j(N--^^^yy-hC (11). Finally, supposing still the condition (10) satisfied, the solution of the equation Mdx + Ndy = will be J3a.-,j(N-^)dy=C.. (12). 2. The practical rule to which the above investigation leads is the following. To solve the equation Mdx + Ndy = when its first mem- ber is an exact differential, integrate Mdx with respect to x, regarding y as constant, and adding, instead of an arbitrary constant, an arbitrary function of y, which must afterwards be determined by the condition that the differential coefficient of the sum with respect to y shall be equal to N. Then that sum equated to an arbitrary constant will be the solution required. Ex. 1. Given (x^ — 4xy — 2y^) dx + (if — 4ixy — 2x^) dy = 0. Here M== x" — ^xy — 2^ and N—y^^ ^xy — 2a;'', whence dy dx ^' and the first member of the given equation is an exact diffe- rential. AET. 2.] OF THE FIEST DEGREE. 51 Nowl |ilf^aj = |--2^'2/-2/a;+^(2/) (1), the arbitrary function (?/) occupying, according to the rule, the place of the constant of integration. To determine (2/), we have <^ I K _ 20,^2' - 22/^^ + <^ (2/)} = 2^' - 4^2/ - ^^ '^ dy\ Whence ^,^ = y\ dy -J' Substituting this value in the second member of (1), and equating the result to an arbitrary constant, we have the solution required. Ex.2. Given --^^A\^ f J - = Q- Here ilf= ,, } ,, , iV=-^ ^ Hence we find dM___-2yi___dN dy {^-\-y'^)^ dx ' To obtain the complete integral we will on this occasion employ directly the general form of solution (12). We have Mdx = log {x +V («' + /)}, /■ dyj' Mdx = -- "" d C Hence iV— -7- I Mdx = 0, so that (12) gives simply log{aj + V(^' + 2/')}=C- 4—2 52 EXACT DIFFERENTIAL EQUATIONS [CH. III. Substituting log C for c, and then freeing the equation from logarithmic signs and from radicals, we have 3. We may in many cases either dispense with the appli- cation of the criterion (1), or greatly simplify its application, by attending to the two following principles, viz, 1st, If Mdx -f Ndy can be divided into two portions, one of which is manifestly an exact differential, it suffices to ascer- tain whether the other is such. 2ndly, If Mdx -i- Ndy, or that portion of it which, according to the above principle^ it may suffice to examine, can be re- solved into two factors, one of which is manifestly the exact differential of a function of x and y, which we will represent by u, then when the other factor is expressible as a function of u, we shall have an expression of the form / (u) du which is necessarily an exact differential. Ex. , Given ^x + ^jj^^ d^+[j- y^if-.^'} ^V = »• This equation may be expressed in the form , , ydx — xd& „ xdx-\-ydy-\-'-r^^ f^ = 0. i^ow, xdx + ydy being an exact differential, it suffices to ex- iidx xdii amine whether the term • . ^ ^ is such also. This term may be expressed in the form of the product y ^ydx-xdy X the second factor of which is the differential of - . If we y make - = It the product assumes the form -rpr ^, which is y ^ ^ ^{L-u) the differential of sin"^z^. I ART. 4.] OF THE FIEST DEGREE. 53 Tlie complete primitive is therefore 2 I 2 2 y 4. The converse form of the property last noticed is of sufficient importance to be stated as a distinct proposition, namely, Prop. II. If U and u be functions of x and y, and Udic be an exact differential, then U will be a function of u. For Udu = Z7-^ dx + U -^ dy. ax dy '^ Hence the second member being an exact differentialwe have by Prop. I. dy \ dx) dx \ dy ^, f, dUdu dU du ^ tnereiore -^ — ^ ^ — ^ = 0. dy ax ax ay Therefore, by the proposition in the first Article of the second Chapter, ZZwill be a function of u. EXERCISES. • 1. {x" + 3^/) dx + {f + Sic'^) dy = 0. •2. {l-\-^dx-2'^dy = ^, \ X J X 2,xdx ■ S. y ■^ (^-1)^^=0- . -, ^ xdy — ydx ^ 4. xdx + ydy -\ ^, — ^-tt— = 0. ^ ^ x'-vy 6. e'' {^:£- ■\-f-V 2x) dx + ^ye'dy = 0. 54 EXERCISES. [CH. III. ' 7. [n cos {nx + my) — m sin (mx + ny)] dx + {m cos {nx + my) — n sin (two? + wy)} dy = 0. 8. Shew, without applying the criterion, that the follow- ing are exact differentials, viz. ^ , xdx + ydy ydx — xdy ^ 1st, J- H 2— — „ = 0. X 2ndly, ^-^ — '^-^ j + 1 — — -^ — 2^ + ^r iydx-xdy). {x'+yf{i-x'-y')^ wiy-^) yy^ ^^ 9. Integrate the above equations. x'^dii ~" CLiix^ dx - 10. Integrate the equation — '-j-^ — ^—^ h x'^'^dx = 0, distinguishing between the different cases which present them- selves according, 1st, as h and c are of the same or of opposite signs ; 2ndly, as a is equal to, or not equal to, 0. 11. Shew by the criterion that the expression is generally an exact differential, and exhibit the functional , . , dM , dN forms which -^r- a-iid -y- assume. ay ax ( 55 ) CHAPTER lY. ON THE INTEGRATING FACTORS OF THE DIFFERENTIAL EQUATION Mdx + Ndy =0. 1. The first member of the equation Mdx + Ndy = not being necessarily an exact differential, analysts have sought to render it such by multiplying the equation by a properly determined factor. Thus the first member of the equation (1 4- 2/^) dx+xydy = is not an exact differential, since it does not satisfy the con- dM dJSr dition -^— = -^f- y t>^^ it becomes an exact differential if the ay ax equation be multiplied by 2x, and its integration, which then becomes possible, leads to the primitive equation The multiplier 2x is termed an integrating factor. We propose in this Chapter to demonstrate that integrating factors of the equation Mdx + Ndy = always exist, to in- vestigate some of their properties and relations, and to shew how in certain cases integrating factors may be discovered. To complete this subject we shall, in the next following Chapter, investigate a partial differential equation, upon the solution of which their general determination depends, and shall examine some of the conditions under which the solu- tion of that equation is possible. 56 ON THE INTEGEATING FACTORS. [CH. IV. 2. To every differential equation of the form Mdx-\-Ndy=-Q, pertains an infini te number of integrating factors, all of which are included under a single functional expression. It has been shewn, Chap. II. Art. 2, that the above equa- tion always involves the existence of a' complete primitive of the form ylr(x,y)=:c (1). Differentiating the last equation, we have d^\r{x,y) ^ df(x,y) dy ^^ dx dy dx The value of -j- determined as a function of x and y from dx ^ d\i this equation must be the same as the value of -^ furnished by the given differential equation expressed in the form ax Hence eliminatinof -~ between these equations we have dx d-^ {x, y) d^\r( x,y) dx Jy fi^\ ' i'^)' M JS' Let ju- be the value of each of these ratios, then As iiM and [juN are therefore the partial differential co- efficients with respect to x and y of the same function i|r {x, y) , the expression ^Mdx + fiNdy will be an exact differential. Thus Mdx •\- Ndy is jlways^ susceptible of being made an exact differential by a factor /x. ART. 3.] ON THE INTEGRATING FACTORS. 57 3. The form of the complete primitive is however without gain or loss of generality susceptible of variation. Thus the primitive a:;^ (1 + 'if) = c, Art. 1, might, without becoming more or less general, be presented in the forms mi[x\l+y')] = c^, log{a^'(l + ?/')} = C2, :^ or in the functional form f{x^O- + 3/^)} = ^y where c, c^, c^ are arbitrary constants. And generally a complete primitive ex- pressed in the form F= c may be expressed also in the form y* ( F) = c, / ( F) denoting any function of V. These variations ^ in the form of the complete primitive imply corresponding i variations in the form of the integrating factor, a special deter- ^ J mination of which has already been given, Art. 1. To investigate the general form under which all such special determinations are included, let us suppose /jl to be a particular integrating factor of Mdx + Ndi/, and let fjuMdx + \xNdy be the exact differential of a function -x^ (^, y). Then representing for the present '^ (cc, ?/) by v, we have fxMdx + jJbNdy — dv. Multiply this equation byj^(v), an arbitrary function of i;; such being, by Art. 4, Chap. III., the general form of a factor which will render the second member an exact differential. We have fif {v) {Mdx 4- Ndy) =/ {v) dv. Now the second member of this equation being an exact dif- ferential the first is so also. As moreover the first member of the above equation can only become an exact differential simultaneously with the second, the factor /x/(v) is the general form of a factor which renders Mdx + Ndy an exact differential. We may express the above result in the following theorem. If lite an integrating factor of the equation Mdx +N'dy= 0, and if v = c le the complete primitive ohtained hy multiplying the equation hy that factor and integrating, then /Jbf(v) will he the typic al for m of all the integrating factors of the equation. Furthermore, /(v) being an arbitrary function of v, the num- ber of such factors is infinite. 58 ON THE INTEGRATING FACTORS. [CH. IV. Ex. The equation [x^y - 2y') dx + Q/'aj - 2x') dy = 0, becomes integrable on multiplying it by the factor ( — ) , the actual solution thus obtained being y X Hence the general form of the integrating factor of the equa- tion is ^ ^ ^ . 2 "•" 2 ^y \y ^ 4. From the typical form of the integrating factor of the equation Mdx + Ndy = 0, it follows that if we know two par- ticular integrating factors of the equation, the solution may be inferred without integration. For fi being one of the factors given, the other must be of the form fJif{v). If we determine their ratio by division and equate the result to an arbitrary constant we shall have which, from what has been said in the preceding Article, is a form of the complete primitive. 5. It has been observed, Art. 1, that the discovery of an integrating factor of the differential equation Mdx + Ndy = generally depends on the solution of another differential equa- tion, but there are some cases in which it presents itself on in- spection. The equation {xy^ +y) dx — xdy — 0, becomes integrable on being multiplied by the factor —^ , and this factor is at once suggested if we place the equation in the form y^xdx + ydx -^ xdy = 0. AET. 6.] SPECIAL DETEEMINATIONS. 59 We could thus, also by inspection, assign the integrating factors of any equation of the form y^dx + ^ {x) [ydx — xdy) = 0, and many other forms will readily suggest themselves. The following analysis will however lead to results of greater generality and importance. Special Determinations of Integrating Factors. 6. Whatever may be the constitution of the functions M and N we have identically Mdx + Ndy = \^{Mx + Ny) (J + ^) + {Mx-Ny) (J * |)} . But ?4^ = cZlog(..),t-f = ^%g). Hence, Mdx-\-Ndy^^{Mx^-Ny)dlogxy + {Mx-Ny)d\^^ (1). The functions Mx + Ny and Mx — Ny appear in the second member of this equation as the coefficients of exact differen- tials. And upon the nature and relations of these functions the inquiry will now depend. Whatever may be the constitution of M and N some one, and only one, of the following cases will present itself. Either the functions Mx + Ny and Mx — Ny will be both identically equal to 0, or one of them will be so and not the other, or neither of them will be identically equal to 0. These cases we will separately consider. 1st. The case of Mx + Ny and Mx — Ny being both iden- tically equal to ma}- be dismissed, as it would involve the supposition that if and N are each identically equal to 0. 60 SPECIAL DETERMINATIONS [CH. IV. 2ndly. Suppose that one of the functions Mx+Ny and Mx — Ny is identically equal to and not the other, and first; let Mx + Ny be identically equal to 0, then (1) becomes Mdx-\-Ndy = l{Mx-Ny)d\og-\ whence dividing by Mx — Ny, Mdx + Ndy ^ ^, x ,^. ~^rj ^=icZlog- (2). Mx -Ny ^ ^ y ^ ^ Now the second member being an exact differential the first member is also one. In this case then Mdx + Ndy is made 1 an exact differential by the factor -^ ^ . By parallel reasoning it follows that if Mx— Ny is identically equal to and not Mx + Ny, an integrating factor of Mdx + Ndy will be - ^— Mx + Ny' And thus we are led to the following theorem. Theorem. If one only of the functions Mx + Ny and Mx — Ny is identically equal to 0, the reciprocal of the other function will he an integrating factor of the equation y Mdx + Ndy = 0. Srdly. Let neither of the functions Mx + Ny and Mx — Ny be identically equal to 0. Then first dividing the funda- mental equation (1) by Mx + Ny, we have Mdx + Ndy . ,, ^Mx-Ny^, x .„, -g^^- = icZlog«^ + i^^^^log- ...(3). Now, by Art. 3, Chap, in., the second member of the above equation becomes an exact differential (its first term Mx Nv . . X being already such) if -j;^ —- is a function of log - ; there- X fore if it is a function of - ; therefore if it is a homogeneous y function of x and y of the degree 0, for the typical form of ART. 6.] OF INTEGRATING FACTORS. 61 sucli a function is ^f-j; therefore, finally, if J/ and iV are homogeneous functions of x and y of a common degree. For let M and N be homogeneous and of the n^"^ degree. Then Mx — Ny and Mx + Ny are each of the degree n-\-l, and Mx — Ny . -^^ zr~- is of the degree 0. Thus J/andiV^beino^ homo£jeneous Mx-\-Ny ^ o & functions of the n*^ degree, the second member, and therefore the first member of (3), is an exact differential. From this conclusion, combined with the previous one, we arrive at the following theoreni. Theorem. The equation Mdx ■\-Ndy = ichen homogeneous // is made integrahle by the factor -^j ^ ,- unless Mx + Ny is 1 . . • identically equal to 0, in which case -yj ^ is an integrating factor. Always then the homogeneous equation Mdx + Ndy = is ** made integrable either by the factor -^ ^, or by the factor Mx — Ny* In the second place, dividing the fundamental equation (1) by Mx — Ny, we have Mdx + Ndy ^fMx + Ny ^\ . Mx-Ny==KM^^^~y'^^'^''^^'^^'^y)'"^^^' of which the second member, and therefore also the first member, becomes an exact differential if -^ ^ is a func- Mx — Ny tion of log xy ; therefore if it is a function of xy ; therefore, finally, if M and N are of the respective forms M=^F,{xy)y, N=F^(xy)xi 62 SPECIAL DETEEMINATIONS [CH. IV. since this supposition would give Mx-Ny~ F,{xy)-F^{xy)' of wliicli the second member is a function of the product xy. Hence the following theorem. Theoeem. The equation Mdx + Ndy = is made integralle hy the factor ^ — — ^ , when M and N are of the respective forms M^F^{xy)y, N=F^{xy)x, unless Mx — Ny is identically equal to 0, in which case -^ ^ is an integrating factor. Or the theorem might be thus expressed. The equation F^ (xy) ydx -f F^ (xy) xdy = is made integrable hy the factor 1 unless we have identically F^ [xy] - F^{xy) = 0, in which case 1 c^y{FA^y)+Kipy)] is an integrating factor. We may, hoAvever, remark that, in the particular case in which F^[xy) — F^{xy) =0, no factor is needed, as the dif- ferential equation may then be expressed in the form F^ (xy) {ydx + xdy) = 0, the first member being manifestly an exact differential. 7. The results of the above investigation may be summed up as follows. ■^^ If either of the functions Mx + Ny, Mx — Ny is identically equal to 0, the reciprocal of the other function is an integrating ART. 7.] OF INTEGRATING FACTORS. 63 factor of Mdx + Ndy = '^lut if neither of these functions is equal to 0, then -j^ ^ is an integrating factor for the equation when homogeneous, and -^ ^ an integrating «f / z factor of the equation when susceptible of expression in the form F^ (xy) ydx + F^ {xy) xdy = 0. Ex. 1. Given x^dxi^{^x''y + 2y^ir^y = 0. This is a homogeneous equation, and its integrating factor according to the rule above given will be 1 Thus we have, as an exact differential equation, x^dx (3^V + ^') dy a;' + 3;z;y + 2?/'"*"i»*+3;:cy + 22/*"" ^■^>'- Referring then to Art. 2, Chap, iii., we have %Jb (JjxJu \Mdx=[ ic' + 3.:cy+2/ 2x x rA dx .a;' +2/ x'^y') Differentiating this expression with respect to y, and com- paring the result with the corresponding term in (1), we find ■ J = 0, whence ^(2/) = const., and we have dy 1 x"- + 22/^^ or a;' + 2/= (7V(^' + /) for the integral required. 64 SPECIAL DETERMINATIONS [CH. IV. Ex. 2. Given {y + xif) dx-\-{x — yx^) dy = 0. This equation may be expressed in the form (1 + xy) ydx + (1 — xy) xdy — 0, Hence its integrating factor, as given by the rule, will be 1 _ 1 Mx — Ny (J. 4- xy) xy — (l — xy) xy n 2 2 * zxy Kejecting the constant \, we have, on multiplying the given 1 equation by ^-^ , X y - l-\- xy T l — xy J ^ — n-^ dso H 2^ dy= 0. xry xy ^ Hence fdx fdx 1 Mdx= -^+ -- = logx-—+^{y), Jxy J X ^ xy ^ ^^' Now Ildv = -^ — — , Hen-ce the complementary function ^ xy^ y

p ^ xy for the integral required. Ex. 3. Given [x^y^ + xf) dx — [x^y + x^y^) dy=0. If we treat this as a homogeneous equation regardless of the implied conditions, we find Mx + Ny 0* The rule however shews that when Mx + Ny is, as in the ART. 9.] OF INTEGRATING FACTORS. 65 above example, identically equal to 0, -^i^ :r^ represents an integrating factor, wliich in the above case will be The equation is thus reduced to X y~' whence w^e find y = c^ as the complete integral. 8. From the theorems of the preceding article others of greater generality may be deduced by transformation. Thus, since the equation FJ^xy) ydx + F^ {xy) xdy = is made inte- grable by the factor — t-^^-t — r ^^-^ — -r , it follows that the equation F^ (uv) vdu + F^ (uv) udv = is made inteOTable by the factor — tptt — n — r^—, — ^ 5 ^ and * ^ uv\F^{uv)-F^{iLv)}' V being any functions of x and y. Hence expressing du in the form -j- dx-\--j- dy, and dv in the form -^ dx-\-^ dy, we see that the equation ■^^^""^^ "" Tx +-^2 W ''^}'^^ + 1^1^'''') "^ Ty ^^^H ^^ ^dy = is made integjrable by the factor — r-rr^ — ^ tti — rr > what- ^ ^ uv[F^[uv)-F^{iiv)] ever functions of x and y are represented by u and v. And, on giving particular forms to these functions,^ particular con- ditions of integration of the equation Mdx + Ndy — present themselves, 9. An integrating factor for homogeneous equations may also be found by the following method, due to Pro fessor S tokes, who first pointed out the necessity of taking account of the B. D. E. 5 06 HOMOGENEOUS EQUATIONS. [CH. TV. case in whicli Mx + Ny is identically equal to 0. {Camhridge Mathematical Jouriial, Yol. iv. p. 241. First Series.) Suppose 31 and N to be homogeneous functions of a; and y of the degree qi. Then we may write ilf=^"<^H N=(^"f{v) (1), where v stands for - . Hence lldx + Ndy = x""^ \v) dx +.cc"'^/r {v) dy ( 2) . But y-=xv, therefore dy — xdv + vdao. Substituting this value of dy in the second member, we have 3Idx + Ndy = ^"{^ {v) + vf (v)] dx + x'^'^'f {v) dv...{S). Two cases here present themselves. First, the constitution of the functions ^ (v) and -v/r [v) may be such that (f> (v) + vyjr (v) may be identically equal to 0. This will happen if Mx + Ny is identically equal to 0, since Mx + Ny = x''''^{<^{v) + vyir'(v)} (4). 1^ this case the equation (3) reduces itself to Md^ + Ndy = ^"""'^/r {v) dv, Mdx + Ndu , / N 7 or ^.^, = ir (v) dv. Now the second member being an exact differential the first is so also, and Mdx + Ndy is therefore made integrable by 1 the factor — , ,v . X Secondly, the constitution of (f> (v) and yjr (v) may be such that ^ (v) + vyjr (v) is not identically equal to 0. And this happens when Mx + Ny is not identically equal to 0. In this case dividing both members of (3) by we have Mdx + Ndy _dx -^ (v) dv GH. IV.] EXEECISES. 67 But the second member being an exact differential tlie first also is such. Now Mdx + Ndy _ Mdx + Ndy , . . x''^^[(i>{y) + vf{v)] ~ Mx + Ny ^ ^ • Here then Mdx -}- Ndy is made integrable by the factor 1 Mx + Ny' Combining these results together, we see that the homo- geneous equation Mdx + Ndy = is made integrable by the factor -^ T^, unless the constitution of M and N is such. Mx-^Nt/' as to make that factor infinite. In the latter case -— r. will h& x^ an integrating factor, n being the degree of M and N. The form of the supplementary integrating factor as given by the above investigation is different from that before ob- tained. The results are however perfectly consistent. For a more complete analysis of the problem which has for its object the discovery of the integrating factors of a homo- geneous equation we must have recourse to the method of the next Chapter. - EXEECISES. • 1. Shew by the application of the theorem of Art. 1^ Chap. II., that the expression o^y -\-x^ + y^+2 {xy — l){x-\- y) is a function of x and y, only as being a function oixy ■\-x + y. • 2. A particular integrating factor of the equation 2xydx + {y^ — Sx^) dy = is ?/"*. Prove this, and deduce another integrating factor by the formula established in Art. 6 for homogeneous equations. * 3. Exhibit the general form under which all the integrat- ing factors of the above equation are comprehended. 5—2 68 EXERCISES. [CH. IV. ' 4. Deduce in like manner the functional expression for all the integrating factors of the equation ' dx dy ^fdx dij\ ^ 0) y \y X ) 5. Obtain integrating factors for the homogeneous equa- tions : . (1) xdy —ydx=\/{x^ -^y"^) dx, (2) {Sy + 10^) dx + {oy + Ix) dy=0: (3) {x'' + 2xy-y^)dx+(y^ + 2xy-x')dy = 0. (4) y + (^2^ + ^^)g = 0. (5) Ix cos - + 2/ sin -J ydx+ I x cos ~ — y sin ~] xdy = 0. \ X XJ \ X Xj Exhibit the corresponding integrals of the above equa- tions. 1 *H 6. The formula ^, — —r^ fails to give an inteOTating Mx + Ny ^ & e, factor for the homogeneous equation — ^ — '—- — 0. What X ~\' y formula ought here to be employed and to what result does it lead ? 7. Determine an integrating factor of each of the equations ' (1) {x^y^ + xy) ydx + (a^y — 1) xdy — 0. > (2) («'^'+ ic'y + xy + 1) ydx + (^'V - x^- ^y 4 1) ^c?2/-= 0. ( 69 ) CHAPTER V. ON THE GENEEAL DETERMINATION OF THE INTEGRATING FACTORS OF THE EQUATION Mdx -\- Ndy =0. 1. Prop. It is required to form a differential equation for determining in the most general manner the integrating fac- tors of the equation Mdx + Ndy = 0. Let fi be any integrating factor of the above equation, then since fiMdx + fjuNdy is by hypothesis an exact differential, we have by Prop. I. Chap. ill. d(fJLN) ^ d(fM]\I) dx dy Hence dx dx dy dy * or, by transposition, ■j^dfjb _ i\T^_ (dM dN\ dx dy \dy dx J • ' which is the equation required. Now this equation involves the partial differential coeffi- cients of fju taken with respect to x and y. It is therefore a partial differential equation. We have not the means of solving it generally, and it will hereafter appear that its general solution would demand a previous general solution of the differential equation Mdx + Ndy — 0, of which fju is the integrating factor. But there are many cases in which we can solve the equation under some restrictive condition or hypothesis, and the form of the solution obtained will always indicate when the supposed condition or hypothesis is legitimate. Therefore or 70 GENERAL DETERMINATION [CH. V. The following are examples of such solutions. 2. Let yu, be a function of one of the variables only, e.g. suppose fjL = (p (x)j then since -J- = 0, we have from (1) dM_dN <^' (ps) _ dy dx ^~(^)~ * N ' ' dM_rm d ^ , , \ dv cix Now if the second member of this equation is a function of x the equation is integrable, and we have rdM_dN log^{a:)=j~^ dx. Whence /^ = 6-' ' (2). "We have seen that the hypothesis. assumed as the basis of the above solution, viz. that the integrating factor /z, is a function of x only, is legitimate when the constitution of the functions M and -^ is such that the expression fdM _dN\ ^ \dy dx ) ' is a function of x only. In this case (2) enables us to deter- mine the value of /a. In like manner the condition under which /x is a function of y only, is dN^dM Tf^ = a function of y only (3), ART. 2.] OF INTEGRATING FACTORS. 71 and the value of [x, on this hypothesis, is H' — ^ \-^j' Ex. Let us inquire whether the equation (3^' + 6xy + St/') dx + (2x^ + Sxy) %- = (5) admits of an integrating factor which is a function of x only. Making M= Sx^ + Qxy + 3/, ^= 2x' + Sxt/, we find dM_dN dy dx _^x-\-^ij— (4ic + %y) _ 1 xV ^lo? + Sxy x^ and this result being a function of x alone, the determination of yL6 as a function of x alone is seen to be possible. From (2) we now find rdx C being an arbitrary constant. Now multiplying (5) by Cx, we have G[{Sx' + Qx'y + Sxy') dx + {2x' + Sx'y) dy] = 0. The first member of this equation remains a complete differ- ential whatever value we assign to C. If we make (7 = 1, and integrate, we find ox"- ^ ^ SxY the integral sought. The student may obtain also the same result by solving (5) as a homogeneous equation. The linear differential equation of the first order % + Py-Q=o (6). P and Q being functions of x, may be solved by the above method. For, reducing it to the form {Py- Q)dx-\-dy = (7), 72 GENERAL DETERMINATION [CH. V. we have M—Fy — Q, N= 1, -whence dM__dN dy dx _ jy N ' which being a function of x we find from (2) fpdx fi = e . Multiplying (7) by the factor thus determined, we have - e*^^''* {Py - Q)dx+ J''^^ dy = 0, the first member of which is now the exact differential of the function y — I e "" Qdx. Equating this expression to an arbitrary constant c, we find y = f^-^{o,^^J'''Qdx] (8), which agrees with the result of Art. 10, Chap. II. 3. Lei it he required to determine the conditions under tvliicli the equation Mdx + Ndy = 0, can he made integrahle by a factor /jl which is a function of the product xy. Representing xy by v and making fi = (j){v), the partial differential equation (1) becoixies ivvwJ-..^'w|-(f-f)^(.)=o, , . dv dv p , whence, smce ^- = ?/, -^ = ^, we nnd dx ^ dy dM_dF ii>[v) Ny-Mx ^^' Thus the condition sought is that the second member of the ART. 3.] OF INTEGRATING FACTORS. i'S above equation be reducible to a function of v alone, i.e. of xy alone. And the* corresponding value of fx is /d^f_dy ^-. ^^-^^^" ...(10). One case in which the above condition is satisfied is the following, viz. F^{xy)ydx^-F^[xy)xdy = (11). Making iI/= F^ (v) y, N= F^ {v) x^ and observing that since dv dv „ T ''^'^' dx = y^ ^ = ^, we find m_dN dy dx ^ F^ (v) + vF^' (v) - F^ (v) - vF^' (v) Ny-Mx '"[F^{i') - l'\{v)} _ F^{v)-F,(v) + v{F :{v)-F:{v)} Ai\iv)-F,(v)] 1 f ;{v)-f:{v) V F,{v)-F,{v)' a function of v alone. Multiplying by dv and integrating, we have 'dM_ dN ^l _Mx ^^ = - log ^ - log [K W - K {')]' Hence, 1 1 v[F^{v)^F,{v)} xy[F^[xy)-F,{^xy)y This accords with a result of Art. 6, Chap. iv. [The above investigation fails when the constitution of the functions M and N is such that we have identically Ny-Mx = 0, An integrating factor for this case has already been found in the preceding Chapter.] 74 GENERAL DETEEMINATIO^ [CH. V. Ex.1. Thus the equation (x^3/^+l)2/<^^ + (a?^j/^—l)^(% = becomes integrable on being multiplied by the factor ^-^, which is found by substituting in the previous expression x^.]f + 1 for F^ {xy), and x^y^ — 1 for F^ {xy). The final solution is Ex. 2. The equation (2^y - y) dx + (2xY -x)dy = 0, does not fall under the type (11), but the values which it furnishes for M and N give dM_ dN dy dx _ 4{v), and observing that dv — y dv _1 dx~ x^ ^ dy £c' ART. 4.] OF INTEGRATI]S"G FACTORS. 75 the partial differential equation (1) becomes -A^f W|-JI/f (.)l=(|^-f )^W ...(12), JdN_dM\ , d>'(v) \dx dy ) whence -vM = — tf vT*^^ * ART. 8.] OF INTEGEATING FACTORS. 85 and assumes on transposition and division the form dx ^ ^{v) ^^ xM ^^^^^^ dv (j) (v) + Vyjr (y) (f> (v) + V^lr (y) ^ ^' Now the reducibility of an equation of this form to a linear form has been established in Chap. IL Art. 11. Under the general form (24) are virtually included some remarkable equations which have been made the subjects of distinct investigations. Thus Jacobi has, by an analysis of a very peculiar character, solved the differential equation (Crelle's Journal, Vol. xxiv.) (A + A'x + A"y) {xdy - ydx) - {B + B'x + B"y) dy Jr{G-^G'x+G"y)dx=Q (26). If, however, we assume in that equation aj = f +a, y = 7}+l3, we can, by a proper determination of the constants a and /5, reduce it to the form (a? + a'r]) {^drj - Tjd^) - [h^ + h'v) dv -h (cf + cv) d^ = 0, which falls under (24). On effecting the substitution in ques- tion the equations for determiaing a and /S will be found to be ol{A + A'ol + A'' 13) - (5 + B'oL + B'^) = 0, -^(^ +^'a + ^"iS) + (7+ 0'a+ (7"/5 = 0. The most convenient mode of solving these equations is to write them in the symmetrical form a p then, equating each of these expressions to X, we find A-X-^A'a.-^ A"^ = 0, B'h{B'-\)oL + B"l3 = 0, 86 GENEEAL DETERMINATION [CH. V. from which eliminating a and /3 we have the cubic equation {a - x) {f - x) ( c" - x) - b'c {a - x) - a" c {e - x) -ab{G"-X)-^a'b"g+a:'bg'=o (27). If a vakie of X be found from this equation, any two equa- tions of the preceding system will give a and /5. 9. The present chapter would be incomplete without some notice of a method which was largely employed by Eulerg That method consisted in assuming ytt to be a function definite in form as respects the variable i/, but involving un- known functions of x as the coefficients of the several powers of 2/. • After the substitution of this form of /^ in the partial differ- ential equation (1 ) , the result is arranged according to the powers of y, and the coefficients of those powers separately equated to 0. This gives a series of simultaneous differential equations for the determination of the unknown functions of x. But for the success of the method it is necessary that the primary assumption for //, should have been chosen with some special fitness to the object proposed. The following is an example. Required the conditions under which the equation Pydx + {y+Q)dy = admits of being made integrable by a factor of the form y' + Bf + Sy' F, Q, B and >S' being functions of x. '> [x) , viz. * (1) (x^ + y''+2x)dx + 2ydy = 0. . (2) {x' + y^)dx-2cc7/dy = 0. Determine these factors and integrate the equations. • 2. The equation 2xy dx + (y^ — Sx^)dy = has an inte- grating factor which is a function of 7/. Determine it, and ■integrate the equation. CH. v.] EXERCISES. 89 3. Find those integrating factors of the equation ydx + (2^/ — x)dy = whicli are homogeneous functions of x and y of the respective degrees and — 2, and from tlie consideration of those factors deduce the complete primitive of the equation. 4. For each of the following equations examine whether there exists an integrating factor fju satisfying the particular condition specified^ and if so determine the factor, and inte- grate the equation. (1) y {x^ + y"^) dx + X (xdy — ydx) = 0, yu. a homogeneous function of the degree — 3. (2) {y^ + axy^) dy — ay^dx + (x + y) {xdy — ydx) = 0, //, as in the previous example. (•^) (2/ — ^) dy + ydx — xd (-) = 0, fi homogeneous of the degree — 1. (4) {x^ + y^ + 1) dx — 2xydy = 0, /jl a function of y^ — x^. (5) (y — Sx'^y^ — 2x^) dx + {2y^ + Sx^y^ — x)dy = 0,fjb3i. func- tion of x^ + y. (6) (x^+x^y+ 2xy—y^—y^)dx-\-(y^+xy'^+2xy-x^—x^)dy=0, fjb a function of the product {1+x) (1 + 3/); (7) (Sy^ — x) dx+ {2y^ — 6xy) dy — 0, fi a function of x-]-y\ 5. The equation y (ocF-^y^) dx + x (xdy — ydx) — has an integrating factor of the form e'^cj) (x^-^y^). Determine it, and, from the comparison of the result with that of (1) Ex. 4, deduce the complete primitive. 6. The linear equation -^ + Py = Q having an integrating factor of the form e-^^^^, deduce a corresponding expression for an integrating factor of the equation 90 . EXEKCISES. [CH. V. 7. Prove that tlie equation CLtXj OjSC where P is any function of x, has an integratiog factor of the form TT-. Lacroix, Tom. ii. p. 278. {y-Ff 8. Deduce a similar expression for an integrating factor dy dP ^ of the equation -^ + ^^ + -y- A P^ = 0. Z6. ^ ax "^ ax ^ 9. Investigate the conditions under which the equation dx y \ where P and Q are functions of a?, can be made integrable by a factor of the form 7 ,., ,,„ , and determine the form affix). ( 91 ) ^>" CHAPTER YI ^-'^ OF SOME EEMAKKABLE EQUATIONS OF THE FIRST ORDER AND DEGREE. 1. There are certain differential equations of the first order and degree, to wMch, in addition to their intrinsic claims upon our notice, some degree of historical interest belongs. Among such, a prominent place is due to two equations which, having been first discussed by the Italian mathema- tician Riccati and by Euler respectively, have from this circumstance derived their names. To these equations, and to some other allied forms, the present Chapter will be devoted. Riccati's equation is usually expressed in the form ^ + 6w^ = c^"^ (1). But as both it and some other equations closely related to it and possessing a distinct interest, may, either immediately or after a slight reduction, be referred to the more general equation • X da; -ay + hy'^^cx'' (2), the discussion of which happens to be much more easy than that of the special equations which are included under it, we shall consider this equation first. To reduce Riccati's equation under the general form (2), it suffices to assume u = - . We find, as the result of this X substitution in (1), dy X dx -y + bf = cx'*' (3), which is seen to be a particular case of (2). 92 OF SOME KEMARKABLE EQUATIONS OF [CH. VI. Of the equation x~- — ay-\-hif = cx^. 2. The discussion upon "which we are entering may be divided into two parts. First, we shall shew that the equa- tion is solvable when n — 2a. Secondly, we shall establish a series of transformations by which a corresponding series of other cases may be reduced to the above. 3. First. The equation x~ — ay -\-hy^ — cx^ is solvable when n = 2a. For, assuming y = x^'v, we find on substitution x""^' ^ + hx'"v' = caj", ax whence, dividing by x^'^, we have ax Now if n = 2a the above becomes x'-^^ + hv' = c, ax whence dv _ dx LA] an equation in which the variables are separated. If we restore to v its value ^ and transpose, this becomes ■ X dy — ayx ax a-i^ _ n. > (a\ ly._^^.. +^ ax-y),. w, ART. 4.] THE FIRST ORDER AND DEGREE. 93 an exact differential equation, of whicli the solution will be _fc\i , Ce-^+l '^'^• Ce~~^~ - 1 , = (-|)V..{.-tMi-j, according as h and c have like or have unlike signs. 4. Secondly. The solution of the equation dx — ay-\- hif — ex"" is always reducible by transformation to the preceding case T n±2a . ... , wnenever — ^ — = ^, a positive integer. For let y = A-] — , y^ being a new variable which is to replace y, and A a constant whose value is yet to be deter- mined. On substitution and arrangement of the terms we have Vi Vi Vi dx ^ ^ • Now let -aA-^ hA^ = 0, then J. = ^ or 0. These values of A we shall employ in succession. '/ "^ a. 5. First. If we assume A — j the above equation becomes Vi Vx yl dx — -v: 2 ' >/ Multiply Id g this equation by --^ and transposing, we have 94 OF SOME EEMAEKABLE EQUATIONS OF [CH. VI. Now this equation is of the same form as the given equation between y and x. The coefficients however differ, in that h , and c have changed their places, and a has become a + n. And this transformation has been effected by the assumption a x^ Hence, if in the transformed equation (6) we make a second assumption a + 71 x^ we shall have as the result a'J'-(« + 2n)y, + 6y/ = C33" (7), h and c again changing places, and a + 71 becoming a + 2n. And the result of i successive transformations of the same series will be to reduce the given equation either to the form or to the form x-y^-{a + in)yi-{-cyl = lx'' (8), x-^- {a-\- in) yi + hyt = ex"" (9), according as the integer i is odd or even. Now by what has been established in Art. 3 the above equations will be integrable if we have 71 = 2 (a + in)y an equation which gives n — '^a 2n ,(10). 6. Secondly. If we assign to A its second value 0, (5) becomes (V X T X X Ui If ^ ~ Vi Vi Vi d^ ART. 7.] THE FIRST ORDER AND DEGREE. 95 Or, multiplying by --\ and transposing, ^^'-(^-«)2/i+C2/,' = ^^" (^^)- Now this equation for y^ differs from the equation (6) ob- tained for y^ in the previous series of transformations only in that a in the coefficient of the second term has become — a. With this change only then that series of transformations may be adopted in the present instance. The change of a into —a in the final condition (10) gives n + ^a _ . as a new condition under which the equation in y is solvable. If i=l this gives n = 2a, the condition first arrived at, and upon which the subsequent researches were based. Collecting these results together we see that the equation fjlj fn -4- 2^ x-~ — ay + hy^ = ex'' is integrahle whenever — ~ — - is a positive integer, tr 7. Let us now examine the form in which the solution is presented. (Yi __ 2cs If —^ — = i, which is the condition arrived at in Art. 5, An * we have the series of transformations a . CO y-i^-> and finally a-{-2n a;" _a+ (/— 1)71 cc" 96 OF SOME REMARKABLE EQUATIONS OF [CH. VI. where h—hoTc, according as i is odd or even; and the effect of these transformations is to reduce the given equation to one or the other of the forms (8) and (9). If in the above expression for y we substitute for y^ its value in terms of y^, in that result again, for y^ its value in terms of ^3, and so on, we find a ic" a + n X e a -\-2n , . , -r- (^)' the last denominator beinsr ^ 1 — . The value of y. must then be determined by the solution of (8) or of (9), these equations being now susceptible of expression as exact differential equations in the forms a^"^'%i - (^ + ^'^) y,^"^'''~'dx ■,,,_,, _ ^ .^. '^-""""%i - (Q^ + in) y,x''^'''-^dx ^g+.-.-y^ _ q .Q^ hy^—cx'' ^ ^* When therefore — ^ — =i a positive integer j the solution of the equation x-j- — ay-^ hy^ = ex"" will he expressed in the form of a continued fraction hy (A), the value of y.. in the last denomi- nator being given hy the solution of the exact differential equa- tion (B) or (C) according as i is odd or even. n "4~ 2^ Secondly, if — ^ — = i, which is the condition arrived at in Art 6, we have the series of transformations n — a x^ ART. 7.] THE FIRST ORDER AND DEGREE. 97 _2n-a ^ ^-^ — h — +^ (''^' where h = h or c, according as ^ is odd or even. From these, eliminating, as before, the intermediate variables y^, yii'"yi~x-> we find ^" J c 27^ - a a;" h Sn — a /-pj\ G ' ' the last denominator being ^ j- -I . In this case, however, the equation for y. formed by changing a into — a in B and C will be di — V — fJii JrOG'''-''~^dx = ^ (E), cyl -hx"" or — — f-^, '^ ■ hic G^a; = (Jb), by, -ex according as i is odd or even. When therefore — ^ = i a positive integer, the solution of x-j- — ay -^hy^ = cx^ is expressed by (D), the value of y, in the last denominator being given by the exact differential equation (E) or (F) according as i is odd or even. Ex. Given x -j^ — y + y"^ = x^', B. D. E. 7 98 OF SOME REMAEKABLE EQUATIONS. [CH. VI. „ „ ^ 1 n + 2a ^ , ., n — 2a - Here w = f , a = l, and as — ^ — = z wniie — ^ = — 1, the formula (D) and (F) must be employed. Assuming therein a = 1, 5 = 1, c = 1, n = ^, 1 — 2, we have 2/ = F= ^ I .........(loy, 2^2 being given by the exact differential equation 1 -. . -2 from which we find iiogC^^:i^J)4.s^*= a. (15). The elimination of y^ between (13) and (15) gives 2^l^^SJ-y ^^.^^ ^ Syxi+Sx^ + y ^ ^ which is the complete primitive. tA/ Ex.2. Given ^ + t^^ = ax This is an example of Riccati's equation. Assuming there- 1 fore u=^,we find x-^ — y + y^ = x^, which is identical with the equation last considered. Substituting therefore in (16) ux for y, we find after reduction % o ^ o i+6^^ = C 17). i ART. 8.] GENERAL OBSERVATIONS. 99 General Observations. 8. The connexion between tlie two conditions for the solution of the equation x-~- — ay + hy^ = ex"", implied by the double sign in the equation — ^ — = i, may otherwise be established as follows. If the differential equation be written in the form ^i^'y{y-i)='^^' ^''\ it becomes evident that it is symmetrical with respect to a a y and y — r* Assume then y — t as a new variable in place of y, and writing y — r^y, y — y'-^j^ the equation becomes dy dx x-^-\-h(y+'^y=cx'' (19), or x-^^ay-\-hy^ = cx'' (20), an equation which differs from the given equation only in that y has become y, and a has changed its sign. Hence the conditions n = ^. — =- and n = ^. — ~ are mutually dependent, and the value of y having been obtained for the former case, its value in the latter will be found by changing therein a into — a, and finally adding j- . It is here also to be noted that instead of beginning with an assumption of the form y^A-\ as in Art. 4, we might X have commenced our reductions by the assumption y = jj , the former of the above being proper for increasing by n, the 7-2 BOOLE'S DIFFERENTIAL EQLLmOXS. Fio.fll. %I T M d M' Fig. 1\' Fio. II A ,'.S T"^ M _X 103 GENERAL OBSERVATIONS. [CH. YI, latter for diminisliing by ti the quantity a. And as the first led directly to the solution (A), so would the second have led directly to the solution (D). Lastly, it may be remarked that each of the above assump- tions is only the inverse of the other. To increase the value of a by 9^ we had to employ the assumption iJi which p-ives &' and this indicates the form of the assumption for the case in which a is to be diminished. Hence also by admitting nega- tive as well as positive values of i, the two forms of solution might be replaced by a single one. 9. We have seen in Art. 1 that Riccati's equation ax is reduced by the assumption u = -io the form x^-y-^ly'' = cx'^^\ Hence the condition for the solution of Riccati's equation, found by substituting in the final theorem of Art. 6, 1 for a and m + 2 for n, will be m + 2±2 2m + 4 ~ h irv) — ^ -J- 2 "^ --2i — [ whence (21), j^'C — J. i being a positive integer. ART. 10.] GENERAL OBSERVATIONS. 101 We may give to the expression for m another form, viz. — 4i . . . m =-^. — ^ ,i admitting of the value together with positive integral values. In order to prove this, let it be observed that two values of m included in (21) are — 4^ , — 4 (i — 1) '"=2731- ^"'^'"= 2i-l • If in the second of these values we change ^ — 1 into i, and therefore i into i -|- 1, a change which merely involves that we interpret i as admitting of the value as well as of posi- tive integral values, we find m = ^r.—- (22). 2^ + 1 ^ ^ When ^ = this gives m = 0, and as this value also results from the first of the expressions for 7?^ on making i = 0, we are permitted in that formula also to regard i as admitting of the same range of values. Hence, combining the two formulae in a siDgle expression, we have -=.TI1 (28), i being 0, or a positive integer. 10. Riccati's equation may also be reduced, and it usually has been reduced, by a series of double transformations, of which the following will serve as an example. The equation being -=--\-hu^ = ex'"', let it = ,— + -^ We have dii _ 1-2 1 du^ dx hx^ x^Uj^ x\^ dx 7 2 1 2 ^ hx x'u. hx^ x^u^ x^al 102 GENERAL OBSERVATIONS. [CH. VI. Substituting these values in the given equation, we have h 1 du. "Whence, 11 _ (.j,'> 4 2 'A 'i J ^ CO U^ Ju U^ (jjX 2,2^1 + c^-+*^/_ 6 = 0. ax ^ In this equation assume x = z'^^'\ then dx dz dx dz ' whence, after substitution and reduction, ^^^-^u^^-^z --' (24), an equation differing from the given equation, as to its coeffi- cients and indices, in that h has been converted into ^ . c into , and m into -; ; but which is still of Riccati's m+ 3 m+ 3' form. The transformation, it will be observed, is a double one, as it affects the independent as well as the dependent variable. — 4?* Now if m be of the form ^. — =- , we find on substitution 2^ — 1 and reduction m + 4 _ — 4 ({ — 1) - m + 3 " 2jJ^iy^ ' Hence, a second double transformation of the same nature as the last will reduce the differential equation to a form in which the index in the second member will become — ttt^ — ?^^ — ^ . And 2 (^ — 2) — 1 thus after a series of i transformations the index is reduced to 0, and the equation becomes solvable by separation of the variables. AET. 11.] GENERAL OBSERVATIONS. 103 To establish another condition of sokition, assume in the given equation u — -,x = 5"'^% then, after substitution and reduction, we have dz ??^ + 1 "^ m + 1 which, by what has preceded, will be solvable if we have ~m+l~"~2i-l' 4^ whence, m = — 2i + l' Combining these results it appears that Biccati's equation is integrable if m = ^. — = , % being or a positive integer. This agrees with (23). It is manifest from the complexity both of the transforma- tions above described and of the results to which they lead, that Riccati's equation is, in its actual form, far less adapted for such transformations than the equation to which it is so easily reduced. 11. Riccati's equation becomes linear on assuming _ 1 dw hw dx ' The transformed equation is ^,-hcx'^w = (25). We shall consider it under this form in a subsequent Chapter. [See Exercise 3 of Chapter xvii.] 104 ; euler's equation. [ch. vi. To Riccati's equation some others of greater generality may be reduced by a change of variables, e.g. the equation ^ + 6^V = c^~ (26), • V' by assuming cc"*"^^ =^U \ : Elders Equation. - } \<(\ It has already appeared that the solution of a differen- tial equation may sometimes be freed from transcendents introduced by integration. An example of this has been afforded in the instance of the equation dx dy __ + V(l-^^) ' V(l-2/') (Chap. II.), the solution of which is capable of being exhibited in an algebraic form, although immediate integration intro- duces the transcendental functions sin~^^, sin~^y. The inquiry is here suggested whether in any other cases the direct inte- gration may be evaded, an inouiry the more important as our means of integration are so limited. Euler succeeded in ob- taining without direct integration the solution of the equation dx dy ^|{^a^rhx^r cx^ -h e^' 4-/^') "^ V (« + % + c/ + ef + //) ^ ' and of some related forms. The result belongs to the theory of the elliptic functions, and may be established independently by the methods which more peculiarly pertain to that theory. But the method by which Euler arrived at that result demands notice here. 12. To integrate the equation dx dy ^{a-\-hx-\-cx^-\-ex^+fx'') ^/(a + by-\- cy'-^ef^rfy") Representing the polynomials a -{-hx -^cx^ + ex^ +fo*> ^-^^d ART. 12.] EULER'S EQUATIOX. 105 a + % + cj/^ + eif +fy'^ by X and Y respectively, we have to integrate '■'' +.^=0 (2). V(X) • V(FJ The ordinary solution of this equation in the sense of Art. 5, Chap. 1. would be but it is our present object to obtain an algebraical relation between x and y without performing the integrations above implied. Let I ^-.-^TT-v = t, then J=V(X),| = -v(r) (3). Also let x-\-y—'p, x — y = q. We shall endeavour to form a differential equation in which p and ^ are dependent variables, and t the independent variable. From (3) we have J=vm-v(r) (4), J=vm+v(n (5), at at =.hq + cpq + i eq (8/ + q') + \fpq {f -f g') ... (6), since the transformations x-\-y =p, x — y = q give x^-y^=pqy x' -f^{x -y) (x^^ ^xy + y') = q (^~^~ ) 108 euler's equation. [CH. VI. x* - f = is^ - f) {x' + f) = Pi (^2-2- Again, from (3) we have d^x _ dsJ{X) ^ dx dJ{X) _ldX df ~ dt " dt dx ^2. dx' df 2 dy ' whence by addition ^_l(dX dY^ ar ~ '2^\dx dy = h + cp + \e{f + i) + \fp{p'-VZf) (7), on effecting the differentiations and transforming as before from X and y to p and q. Multiplying (7) by q, and from the result subtracting (6), we have U ^+2 Jj. ^n ~ o ~JJt"± df dt dt 2 _q Therefore 2ie + 2/p). 2 d^p 2 dp do ^j, (f df q' dt dt -^^ dp Now multiplying both sides by -j , ^dpd^ ^d^i dt df {dp\^ dt . .dp . . -? — (iJ^i5-=(''+2-^^)di (')' from which, each member being an exact differential, we have on integration G being an arbitrary constant. ART. 13.] euler's equation. 107 Hence -^=^\l{^-^^P+ff)' Therefore by (4) V(X) - V( r) = (^ - y) ^/{G +e{x+y) +/(a + yf] . . .(9), the integral reqTiired. The student may apply the same process of transformation and reduction to the equation dx dy *J{a + bx + ex' + ex^ -\-fx^) \/[a + hy + cf + ef +fy') = (10). The resulting integral will be ^{X) + J(Y)^{x-y)^{G + e{x + y)+f{x + yr]...{l\). 13. It will probably appear that there is something arbi- trary in the mode in which, in the above investigation, the final differential equation (8) between j?, q, and t, upon which the solution of the problem depends, is formed. The analysis which is subjoined may throw some light upon its real nature, and shew of what general theorem that equation constitutes an expression. Prop. Whatever may be the form of the function (^ [x), the following theorem of development holds good, viz. 4> (y) - 9 W = A {f (y) + *' Wl (y-^) +A,{r'{y)+ri^)}{y-^y + A,{cj>^(y) + cl,^(x)}iy-xy-t&c....(12), wherein A , A^, A^, &c. are the coefficients of the successive e^'-l . powers of x, in the development of the function ~ — in a series of the form A^x + A^x^ -!- A^x^ + &c. 108 EULER'S equation. [CH. VI. For let y = X-]- h, then, employing a well-known symbolical form of Taylor's theorem, {x) = 4> (^ + 7?) - ^ (x) 7 ^ hi {e '^ + 1)^ {x) ^{x + h) + (j) (^) (13), where A^, A^, &c. have the series of values above described. Hence, performing the differentiations and replacing x + Jt by y, and h hj y — x, we have ^ (2/) - ^ («) = A, {W1 V{0(2/)} '^^^^ in which (^{x)=a + hx+ ex"" + ex"" +/«^ (17), <\>{y)==a + hy + cy'+eif-\-fif (18). Representing either member of (16) by dt and assuming t as an independent variable, substitute the values hence deter- mined for (/) (x), cf>' (x), Ps} '" Pn T^sii^o supposed to be determined as known func- tions of X and y. And it is now manifest that any relation between x and 3/ which makes either one or more than one of the factors of the first member to vanish, will be a solution of the equation, and that no relation between x and y not pos- sessinof this character will be such. Hence if we solve the separate equations dx ^' ' dx ^'^ ^' dx ^'- ^^^' any one of the solutions obtained will be a solution of (2), since it will make one of its factors to vanish. And if we express the different solutions thus obtained, each with its arbitrary constant annexed, in the forms any product of two or more of these equations will also be a solution of (2), since it will cause two or more of its factors to vanish. Ex. Given the differential equation (IT-«V=0 (4). Here the component equations are dx dy ay = 0, dx-^'^y^^'^ and their respective solutions are logy — ax — Cj = : (5), log y -\- ax— c^ =0 (6). Either of these equations is a solution of the given equation, and so is their product {logy-ax-c^) {logy + ax-c.;)=0 (7). ART. 2.] ORDER, BUT NOT OF THE FIRST DEGREE. 115 2. And here two important questions are suggested. _First, how is it that two arbitrary constants present themselves in the solution of an equation of the first order ? Secondly, is it possible to express with equal generality the solution of the equation by a primitive containing a single arbitrary constant in accordance with what has been said of the genesis of differential equations of the first order, Chap. i. Art. 6 ? These are connected questions, and they will be answered together. The equation (7) implies that y admits of two values each ] involving an arbitrary constant, but it does not imply that y admits of a value involving two arbitrary constants. The , component factors of the solution separately equated to 0, as in (5) and (6), give respectively y^C/% y=C,e-"^ (8), each of which involves one arbitrary constant only, and each of which corresponds to a single factor of the given differential equation. The true canon is, not that a general solution of an equation of the first order can involve only one arbitrary constant in its expression, but that each value of y which such a solution establishes involves in its expression only a single a rbitrary constant. At the same time there is a real sense in which it remains true that every differential equation of the first order, what- ever its degree may be, implies the existence of a complete primitive involving a single arbitrary constant, and there is a real sense in which such primitive constitutes the general solution of the differential equation. To reconcile these seem- ing contradictions I shall shew that if we suppose the arbi- trary constants c^ and c^ in (7) identical, and accordingly replace each of them by c, we shall have an equation which will be, first the true primitive of (4), in that it will generate that equation by differentiation and the elimination of c, secondly its general solution, in that no particular relation is deducible from the solution (7) involving two arbitrary con- stants which may not also, by the use of a lawful freedom of interpretation, be derived from it. 8—2 116 DIFFERENTIAL EQUATIONS OF THE FIRST [CH. VII. Thus replacing c^ and c^ by c, we have (log 3/— ax — c) (logy + aw — c) — (9), whence (log yY - a^x^ — 2c log ?/ + c^ = 0. dv Differentiating, and representing -r^ by ^, 21og'?/~-2a'a?-2c2 = 0, ^i^y y , a^xy , whence c = ~ + log y. Substituting this value in (9), we have fa^xii \ fa^xy \ ^ ( —^ —ax\\ — " -f aic 1 = 0, which reduces to aV (ay -/) = 0. Or, rejecting the factor aV which does not contain _p, and replacing _p by ^, 'dy\ m-y=^' the differential equation given. Thus (9) is its complete primitive. Again, that solution is general. The two relations between y and x which it furnishes are y = Ce", y = Ce-' (10), and these differ in expression from (8) only in that the arbi- trary constant is here supposed to be the same in one as in the other, but as it is arbitrary and admits of any value, there is no single relation implied in (8) which is not also implied in (10). And it is in this sense that the generality of the solution is affirmed. [See the Supplementary Volume, Chapter xx. Art. 1.] ART. 3.] ORDER, BUT NOT OF THE FIRST DEGREE. 117 8. These illustrations will prepare tlie way for the de- monstration of the general theorem which they exemplify. Theorem. If the differential equation of the first order and n^^' degree he resolved into its component equations dx ^' ' dx ^' ^' dx P^^ ' and if the complete primitives of these equations are 1\ = c^ y V^ = c^,... V^ = c^^, then the complete primitive of the given equation will he (P;-c)(F,-c)...(F,.-c)=0. Let us first examine the case in which the proposed diffe- rential equation is of the second degree, and therefore express- ible in the form ( -— —pA [j^~V^] — ^' Suppose that the integral \\ = c^ is derived from the equation -f — p^ — ^ ^J means of an integrating factor jjl^ . Then dV^ = fij^l-j — ^^ j dx. In like manner we shall have dV^ — I^A~j^ "i^sj ^■^- Now taking the equation (F,-c)(7,-c) = (11) as a primitive, we have, on differentiating with respect to x and y, [r,-c)dv,+ {V,-c)dv,^o (12). Therefore c= 'aV^ + dV T' whence ^ _^^ (F, - FJ <^F. '^^ ■ dV^ + dV^ • 118 DIFFERENTIAL EQUATIONS OF THE FIRST [CH. VII. Substituting these values in (11), we have (F,-F,)^^F,^F, = (13), which gives ( F, - V,y fi,fi, {^ -j>^ ( J ~ ^2) = ^ • • • (^^)- And this, on rejecting the factor (F^ — V^YfJi^jui^ which does not contain any differential coefficients, becomes identical with the given differential equation. Hence ( F^ — c) ( Fg — c) = is the complete primitive of that equation. To generalize this particular demonstration it would be necessary to eliminate c between the equation (F,-c)(F,-c)...(F„-c) = (15), and the equation thence derived by differentiation with re- spect to X and y. The ordinary process of elimination, as exemplified above in the particular case in which n = 2, would be complex, but the result may be determined without dif- ficulty by logical considerations. It will suffice for this pur- pose to consider the case in which n = S. We have then as the supposed primitive iv,-c){r,-c)ir,-c) = o (16), and as the derived equation (n-o)(n-o)(f4f J) .(F,-o)(F.-o)(f.fJ) + (F.-c)(F.-c)(5 + f |)=0 (17). Now (16) implies that some one at least of the equations F^_c = 0, F,-c = 0, F3-c = 0, is satisfied. ART. 3.] ORDER, BUT NOT OF THE FIRST DEGREE. 119 If tlie first of these equations is satisfied we have c= V^y and substituting this value in (17) there results {r.-v^){y.-y.)'iy^=o (18)- If the second equation of the system is satisfied we have in like manner iV.-V.)iy.-V.)dr, = (19). If the third equation of the system is satisfied we hpuve {V,-r,)(v,-v^)dr^ = o. ..(20). Hence the existence of (16) as primitive supposes the exist- ence of some one at least of the equations (18), (19), (20), and therefore of the equation (n- v^fi^s- v^nK- Kydv,dr,dr,^o (2i), which is formed by multiplying those equations together. Conversely the supposition that the equation (21) is true, involves the supposition that one at least of the equations (18), (19), (20) is true. The equation (21) is therefore equivalent to the result which ordinary elimination applied to (16) and (17) would give. The same process of reasoning applied to the more general equation (15) as supposed primitive, would lead to a result of the form Kdf\dV^...dV^ = (22), K being the product of the squares of the differences of V V ... V. On comparison with (13) we see that in the particular case of 72 = 2, this is not only equivalent to but identical with the result of ordinary elimination in that case. And this identity of form, though it is not necessary to our present purpose to establish it, might be demonstrated generally. Now dV, = ^1 (^ - Pi) dx, ^ ^2 = /^2 (^ - P,) ^^^^ &c. 120 DIFFERENTIAL EQUATIONS OF THE FIRST [CH. VII. Hence (22) gives or, rejecting the factor KiJb^fjb^...iJb^, which does not contain differential coefficients, (i-^')(J--p^)-(|--P")=^- Of this equation it has therefore been shewn, as was required, that {V^ — c) ( 1^2 ~ ^) ••• ( ^« ~ ^) ~ ^ constitutes the complete primitive. [See the Supplementary Volume, Chapter XX. Art. 2.] Ex. Given ("^^-^=0 (1). Here the component equations are ax \xj ax \xj and their respective integrals are y-c^-2^{ax) = , '(2), 2/-C2 + 2VM = (3). Replacing both constants by c and multiplying the equations together, we have {y — cY — 4tax = (4), as the complete primitive. Now this primitive represents a series of parabolas, the parameters of which are constant and equal to 4a, and the axes of which are parallel to the axis of cc ; but the ver- tices of which are situated at different points of the axis of y, corresponding to the different values which may be given to the arbitrary constant c. Of these parabolas the equations (2) and (3), which may be written in the more usual forms y-c^ = 2^/{ax), y - c^==-2\/{ax), AET. 4.] OKDEK, BUT NOT OF THE FIRST DEGREE. 121 represent respectively the positive and the negative branches, while the equation {y-c^^2^{ax)][y-c,+ 2^J{ax)} = (5), represents the terms which would be found by taking one positive and one negative branch, hut not necessarily from the same parabola. Thus there is no portion of the loci re- presented by the apparently more general solution (5), which is not also represented by the complete primitive (4). The defect of generality, if as such it is to be regarded, consists in this that while each branch of every curve in the series is represented, those branches which belong to the same curve are paired together. [On the subject discussed in the first three Articles of the present Chapter the student may consult a paper by Pro- fessor De Morgan entitled On the question, What is the solu- tion of a Differential Equation ? The paper is published in the Camhridge Philosophical Transactions, Yol. x.] 4. There are certain cases in which differential equations of the first order can be solved without the resolution of the first member into its component factors. Of these the most important are the following. 1st. When the given equation contains only one of the variables x and y in addition to -— , being either of the form ^(.,J) = 0. oroftheform^(y,J)=0. 2ndly. When, involving x and y only in the first degree, " it is expressible in the form xj> {p) + yyjr (p) = X (p), where p = ^- Srdly. When the equation is homogeneous with respect to X and y. These cases we shall consider separately. 122 DIFFERENTIAL EQUATIONS OF THE FIRST [CH. VII. Equations involving only one of the variables x and y with -i- . ax ^- 5. In this case if, representing ;t^ by p, and regarding p as a new variable, we form a differential equation between p and the variable which does not enter into the original equation, and integrate the equation thus formed, the elimina- tion of p) between the resulting integral and the original ' equation will give the complete primitive required. For it will express a relation between x, y, and the arbitrary con- stant introduced by integration. Thus if from the equation F{x,p) = we deduce x =f (p), then, since dy —pdx, we have therefore y—\pf'{p)dp-{-c .....(1). After the integration here implied y will be expressed as a function of p and c, and between that result and the original equation p must be eliminated. In like manner, if from F {y,p)—0 we deduce y=f{p), the equation dy = pdx gives /'(_/9) dp —pdx, whence dx — - — -^ dp, whence =jm#+, (2), X between which (after the integration has been performed) and the original equation, p must be eliminated. But these methods, though always permissible, are only advantageous when it is more easy to solve the given equa- tion, with resjDect to the variable x or y which it involves, than with respect io p. ■ ART. 5.] OEDER, BUT NOT OF THE FIRST DEGREE. 123 Ex. 1. Given a? = 1 + j/. Here dy =pdx =p x '^p^dp = ^p^dp ; 0,^4 therefore V ~a^ '^ ^ (•^) • Now as tlie original equation gives _p = (^ — l)^ the com- plete primitive found by substitution of this valae in (3) will be ^=4'^^-l)*+^ W. and it would be directly obtained in this form by integrating the original equation reduced by algebraic solution to the form dx ^ This example illustrates the process but not its advantages. Ex. 2. Giveif^ =l+p-\- p^. Here dy =pdx =pdp + ^p^dp ; therefore 2/=9+4~ + <^ (fi), between which and the original equation^ must be eliminated. We may do this so as to obtain the final equation between x and 2/ in a rational form ; but, if this object is not deemed im- portant, we may, by the solution of a quadratic, determine p from (5) and substitute its value in the given equation. Ex.3. Given2/ = / + 2p^ Here since pdx = dy we have dx — -dy — 2dp + Qpdp ; therefore x = 2p-\- Sp^ + c. 124^ DIFFERENTIAL EQUATIONS OF THE FIRbT [CH. VII. From this equation we find P--^-i -' G being an arbitrary constan^j; introduced in tlie place of 1'— 3c ; and y will be found by substituting this value of j^ i^ the original equation. Equations in vjhich x and y are involved only in the first degree, the typical form being xcj) [p) + y^ (p) =xiP)' 6. Any equation of the above class may be reduced to a linear differential equation between x and p, after the solution of which^ p must be eliminated. The reduced equation is found by differentiating the given equation and then eliminating, if necessary, the variable y. It may happen that such elimination is unnecessary, y disappear- inof throuofh differentiation. Ex. Let us apply this method to the equation y=xp+f{p) (1), usually termed Clairaut's equatior^.^ Differentiating, we have dp J,, . . dp ' dp whence {oo +f (p)] ^ = 0. Now this is resolvable into the two equations, ^Mur- x+f{p) = (2), ^ = (3;. ax The second of these, which alone contains differentials of the new variables cc and p, is the true differential equation between X and^. ART. 6.] ORDER, BUT NOT OF THE FIRST DEGREE. 125 Integrating it we have ^ = c, and substituting this value of ^ in (1), y=cx+f(c) ".. (4), which is the complete primitive required. But what relation does the rejected equation (2) bear to the given differential equation (1), and what relation to its complete primitive just obtained ? If we eliminate jp between (1) and (2) we obtain anew rela- tion between x and y not included in the complete primitive already found, i. e. not deducible from that primitive by assigning a particular value to its arbitrary constant, and yet satisfying the same differential equation, and, as we shall hereafter see, connected in a remarkable manner with the com- plete primitive. Such a relation between x and y is called a singular s olution. We shall enter more fully into the theory of singular solutions in a distinct Chapter, but the following example will throw some light upon their nature, as well as illustrate the process above described. Ex. Given y — xp-i — . Here differentiating we have „ _ / m\ dp \ p^J dx ' From the equation -^ = 0, we have p = c, whence 2/ = cx + - (o), the complete primitive. From the equation w — r^ = 0,we have 126 DIFFERENTIAL EQUATIONS OF THE FIRST [CH. VII. and this value substituted in the original equation gives, after freeing the result from radical signs, y^ = ^^mx (6), the singular solution. Here the singular solution (6) is the equation of a parabola whose parameter is 4m, and the complete primitive (5) is the well-known equation of that tangent to the same parabola which makes with the axis of x an angle whose trigonometri- cal tangent is c. "^ Now, for the infinitesimal element in which the curve and dn its tangent coincide, the values of x, y, and ~ are the same in both. And thus it is that the algebraic equations of the curve and of its tangent satisfy the same differential equation of the first order. On the other hand, if (5) be regarded as the general equa- tion of a system of straight lines, each straight line in that system being determined by giving a special value to c in the equation, the envelop or boundary curve of the system will be determined by (6). Here the singular solution is presented as the equation of the envelop of the system of' lines defined by the complete primitive. 7. In the second place let us consider the more general equation Differentiating, we have whence \v -f{m ^ -f ip) ^ = 0' {p)> dx f{p) ^'(p) dp p-f{p) p-f(p}' ART. 7.] OEDER, BUT NOT OF THE FIRST DEGREE. 127 a linear equation of the first order by which oo may be deter- mined as a function of ^. The elimination of p between the resulting equation and the given one will give the complete primitive. The typical equation may be reduced to the above form by dividing by -v/r (p), but it may also be treated independently by direct ditferentiatioD. Instead however of forming a differential equation between X and 2^, we may form a differential equation between i/ and p. Or, with greater generality, representing any proposed function ofp by t, we may form a differential equation be- tween either of the primitive variables and t Such a diffe- rential equation will necessarily be linear with respect to the primitive variable retained, and its solution must of course be followed by the elimination of t. And this general procedure, more fully to be exemplified when we come to treat of some of the inve rse problems o f Geometry an d of Optics, is often attendecTwith signal advantage. Ex. Given x-V yp — ap^. We shall reduce this to a differential equation between x and p. Differentiating, we have then eliminating y by means of the given equation, we have ^ \ p J dx -^ dx' which may be reduced to the linear form dx X ap dp~p{i+f)^r+f' its integral being 128 DIFFERENTIAL EQUATIONS OF THE FIRST ORDER. [CH. VII. If in this equation we substitute for p its value in terms of x and y furnished by the given equation, i.e. if we make _y ± \/[y^ + ^ax) we shall be in possession of the complete primitive. Had we chosen to form a differential equation between yl and p, we should have, on differentiating the given equation-' while regarding y as the independent variable, dy ^ ^ dy dy' dx 1 whence, replacing -^ by - and reducing, ay jp dy jp_ 2ap^ dp) 1+^/*^ 1 +y therefore on integration 2/ = ^(l+^^) [^+^^i^V(^+/)-^^Qg{^^ + V(l+p^)}], from which, as before, p must be eliminated. The final results are of course identical. Homogeneous Equations of the first order. 8. Equations which are homogeneous with respect to x and y may be prepared for solution by assuming y = vx. The typical form of such equations is ^"•^g. i')=0 (1). Assuming then - = v, and dividing by ^", we have X 't'{v,p) = (2). ART. 8.] HOMOGENEOUS EQUATIONS. 129 If we can solve this equation with respect to p, we have p=f{v). But, since y = xv, dv Thus the transformed equation becomes dv ... , dv dx ^ whence ;:— ^ + _ = 0, v-f{v) X an equation in which the variables are separated, and in the integral of which it will only remain to substitute for v its value - . X But if it be more easy to solve (2) with respect to v than with respect to p, and if the result be then restoring to v its value - , we have Avhich is a particular case of the equation of the previous section. Hence differentiating, we have from which results dx ^ f (p) dp ^^ E. D. E. 9 130 HOMOGENEOUS EQUATIONS. [CH. VII. an eqiiation in which the variables x and p are separated. Between the integral of this equation and the given equation 2J must be eliminated, and the relation betv/een x and y which results will be the complete primitive. Ex. Given yp + nx = \/(^y" + nx^) \I{1 -\- p^). Assuming y = vx, we have vp-\-n = a/(v^ + n) V(l + p'), the solution of which with respect to p gives P=«+A/('^)v(r + «). „ . dv But p = X -r + V, ax Therefore * £ = ± y^ f^) V("^ + ")> (^v I (n — V\ dx ^{y'' -\- n) ~ y \ n J X ' Integrating, we have log [v 4- V(v' + ^)} = ± a/C"^) ^og X -V C', therefore -y -f isj{if ■\-')i)=cx "^ ^ » ^ ^ or, replacing v by - , the com^plete primitive. AET. 9.] equatio:n's solvable by differentiation^. 131 Equations solvable hy differentiation. 9. A remarkable class of equations, the theory of which has been fully discussed by Lagrange, deserves attention. It has been shewn. Chap. I. Art. 9, that if two differential equations of the first order, each involving a distinct arbi- trary constant, give rise to the same differential equation of the second order, they are derived from a common primitive involving both the arbitrary constants in question. Let us suppose these differential equations of the first order to be reduced to the forms dy\ and let the primitive obtained by the elimination of -y- be (a?, y, a, h) = 0. Lagrange has then observed that if we have any differential equation of the first order of the form ^{'^'i^'^'S)' t(-.2/.J)} = (3X its complete primitive will still be ^ {x, y, a, h) = 0, but with the condition that a and b are no longer independent con- stants, but are connected by the relation F(a,b) = 0. This is an obvious truth. For as, by hypothesis, the sup- posed primitive ^ {x, y, a, b) =0 gives it will convert (3) into F (a, h) = 0, and will therefore satisfy that equation if a and b are connected by the relation F{a,b) = 0. 9—2 132 EQUATIONS SOLVABLE BY DIFFERENTIATION. [CH. VII. Moreover it contains virtually only one arbitrary constant, for the relation F {a, h) = permits us to determiue 6 as a function of a. Hence it will constitute the complete primitive ' of (3). See also Chap. I. Art. 10. This result may be expressed in the following theorem. If any differential equation of the first order he expressible in the form Fi4,,f)=0 (4), dv where and 'yfr are functions of x^ y, -— , such that the dif- ferential equations (f) = a, ■f = h, are derivable from a single primitive involving a and h as arbitrary constants, the solution of the given differential equa- tion will be found by limiting that primitive by the condition F(a,h)=0, so as actually or virtually to eliminate one of the arbitrary constants. Ex. Suppose that the given equation is Now the differential equations of the first order are derivable from a common primitive; for, on differen- tiating them, we have respectively ^+sy+2/3=o, dy ART. 9.] EQUATIONS SOLVABLE BY DIFFEEENTIATION. 13o and these agree as differential equations of the second order, Chap. I. Art. 9. That common primitive, found by elimi- nating -^ between (2) and (3), is Hence the primitive of the given equation is f + (.x-af = {f{a)f (4). We might also proceed as in the solution of Clairaut's equation. Differentiating the given equation, we have dy The second factor, which alone involves -7-^ , equated to 0, gives on integration the primitive - ■ 'if-vix- of = ¥, as will be seen in Chap. x. Art. 1, in which the relation be- tween h and a remains to be determined as before. The first factor equated to constitutes the differential equation of the dif singidar solution, which will be obtained by eliminating -j-^ between that equation and the equation given. Clairaut's equation belongs to the above class. We may express it in the form ^ ax "^ \axj Now the differential equations dij _ dx 134 EXAMPLES OF TRANSFOEMATION. [CH. VII. generate the same differential equation of the second order and are derivable from the same primitive y — hx -\- a. Exami^les of Transformation. 10. Well-chosen transformations facilitate much the solu- tion of differential equations of the first order. Ex. 1. Given -~- — ^ = f(x^ + v^)^. Lacroix, Tom. Ii. p. 292. Assuming x = r qo^6, y = r sin 6, we have T:=f{r), -r' whence Consequently ^^-'£J Ar rV[r^-f /(r)l'| dd f{r) /, _ I f{r)dr ^ As —- — ^ is the expression for the lens^th of the per- .V(l+P) . . pendicular let fall from the origin upon the tangent to a curve, the above is the solution of the problem which pro- poses to determine the equation of a curve in which that perpendicular is a given function of the distance of the point of contact from the origin. ART. 10.] EXi3IPLES OF TRANSFORMATION. 135 By the same transformation we may solve tlie equation Ex. 2. Given ($)'= ^^ -{- %^. To render tlie above equation homogeneous if possible let y = z'^ ] we find \ ax) This will be homogeneous with respect to z and a*, if we have h (w — 1) = a = np, equations from which we deduce n = the former of which expresses a condition between the indices of the given equation, the latter the value which must be given to n when that condition is satisfied. It appears then that the equation 'iy-A.^Bf, can be rendered homogeneous by the assumption y — z^. If the more general transformation y — z"^^ os — f^, which seems at first sight to put us in possession of two disposable constants, be employed, the necessity for the fulfilment of the same condition between a, /3, and k, will not be evaded, the ratio of the constants m and n, not their absolute values, proving to be alone available. Ex. 3. The equation of the projection on the plane xy of the lines of curvature of the ellipsoid is 136 EXERCISES. , [CH. VII. Assuming ^ = s, y^ — t, the equation is reduced to one of Clairaut's form, Art. 6. Its solution is Tip y ^^ AG-vV The equation may also, without preliminary transformation, be integrated by Lagrange's method, Art. 9. We may ex- press it in the form A^-f^B^-^-^^^... (2), where ^ = — , '^ = 2/^ — ypx. OS Now ^p z=a, y^ — yjpx — h, are derived from a common primitive 'if — aa? = h. The solu- tion of (2) will therefore be, 'if — ax^ = h with the connecting relation between the constants, Aah + Ba + h = 0, And this will be found to agree with the previous result EXERCISES. The following examples are cliiefjy in illustration of Arts, 1, 2, 3, 5. 1, ax/ \axj 2. ('fV--' = 0. \dxj X ^ (dy^ _\ — X \dxj X OH. VII.] EXEKCISES. 1S7 \dxj y ax _ dy ^ (dy^\ dx \dx) ' (i. x^a'^ + hi'^f). ax \axj dx xy \ \dx) j 11. 1 _,_(^<^3/y_ fa+c^)' The following examples are intended to illustrate Art. 6. The singular solutions as well as the complete primitives arc to be determined. 12. y^x^y~^^-y^-{^ dx dx \dxj The following examples are in illustration of Arts. 7 and S. 138 15. 16. 17. EXERCISES. [CH. VII. ^ ax <^2/ _ ^ /^y 1 + dij dx dx \dxj x + y dy 1 + dy dx "'^ 'Y [ '' ' \dx/ The following examples are in illustration of Art. 9. 18. 41 = ../{/ -,'(!)]. rr=/^^ + 19. 2J- 20 v-'>x^ = f\x(^'] ^ dx -^ \ \dx) \ ' dy dx dx] ) J dy ^ dx y-x f^% \i+-^ dx ^dy^ dx ( 139 ) CHAPTER YIII. r~ ox THE SINGULAR SOLUTIONS OF DIFFEEENTIAL EQUATIONS OF THE FIRST ORDER. 1. In the largest sense which has been given to the term, a singular solution of a differential equation is a relation between the variables which reduces the two members of the equation to an identity, but which is not included in the complete primitive. In this sense, the relation obtained by equating to some common algebraic factor of the terms of the equation might claim to be called a singular solution. But, in a juster and more restricted sense, a singular solution of a differential equation is a relation between x and t/, which satisfies the differential equation hy means of the values which it gives to the differential coefficients -j- , -—^ , &c., but is not included in the complete primitive. In this sense the equa- tion x^ + y^ = n^, is a singular solution of the differential equation of the first order It reduces the members of that equation to an identity, but not by causing any algebraic factor of them both to vanish. At the same time it is not included in the complete primitive y — cx = n a/(1 + c^). And this is , the juster definition, because that which is essential in the singular solution is thus in a direct manner connected with that which is essential in the differential equation. De£ Chap. i. // 140 ON THE SINGULAR SOLUTIONS OF [CH. VIIT. When it is said that a singular solution of a differential equation is not included in tlie complete primitive, it is meant that it is not deducible from that primitive by giving to the arbitrary constant c a particular constant value. But although a singular solution is not included in the complete primitive, it is still imj)lied by it. U]3on the possibility of satisfying a differential equation by an infinite number of particular equa- tions, each formed by the particular determination of an arbitrary constant, rests the possibility of satisfying it by another equation, to the formation of which each particular solution has contributed an element. We have seen in Chap. VIL how a singular solution, as representing the envelope of the loci defined by the series of particular solu- tions, possesses a differential element common with each of them. We shall now see that this property is not accidental — that ifc is intimately connected with the definition of a singular solution. It is important that the two marks, positive and negative, by the union of which a singular solution of a differential equation of the first order is characterized, and by the expres- sion of which its definition is formed, should be clearly appre- hended. 1st. It mus.t sfive the same value of -^ in terms of ^ ^ dx and y, as the differential equation itself does. This is its positive mark, a mark which it possesses in common with the complete prhnitive, and with each included particular primi- tive. 2ndly. It must not be included in the complete primitive. This is its negative mark. Upon the analytical expression of these characters the entire theory of this class of solutions depends. Among the different objects to which that theory has 'reference, the two following are the most important. 1st. The derivation of the singular solution from the complete primitive. 2ndly. The deduction of the singular solution from the differ- ential equation without the previous knowledge of the com- plete primitive. The theory of the latter process is so de- pendent upon that of the former that it is necessary to consider them in the order above stated. [Important additions to the present Chapter are given in the Supplementary Volume, Chapter xxi.] ART. 2.] DIFFESENTIAL EQUATIONS OF THE FIEST ORDER. 141 Derivation of the singular solution from the complete primitive. 2. The complete primitive of a differential equation of the first order^ whatever may be the degree of the equation, is of the form ^ (^, y, c) = 0. If we give to c a particular constant value in this equation we obtain a particular primitive. If we give to c a variable value by making it a function of oo, or of y, or of both, we, as will immediately be shewn, convert the equation into any desired relation between x and y. We propose then to detei;- mine c as variable, but as so varying that the resulting relation between x and y sliall continue to satisfy the differ- ential equation. The general effect of the conversion of c into a function of X or of y must first be considered. Prop. i. A primitive equation (j) {w, y, c) = may,hy the conversion of c into a function of x, he transformed into any desired equation containing x and y together, or y alone, hut 7iot into an equation involving x without y. Let the desired result of transformation be 'f (^> y) =0. oxx {y) = 0, involving y at least. Combining either of these equations with the primitive we can eliminate y, and so obtain a rela- tion between x and c which will determine c as the function of X required. It is evident however that the conversion of c into a func- tion of X could not convert the primitive into an equation not involving y. For a variable cannot be eliminated from an equation, except by the aid of another equation which contains that variable. 142 DERIVATION OF THE SINGULAR SOLUTION [CH. VIII. Similarly the conversion of c into a function of y would enable us to convert the given primitive into any desired equation involving, of the two variables, at least x. Ex. Let it be required to convert the equation y = cx into x^ -\-y^ = 1, by the conversion of c into a function of x. Eliminating y from the given and the proposed equation, we have _y j^ 2 ,2 -1 Ji/ "t~ w X -^ X. , , Vri-a;') whence c = — ^ . X This value of c substituted in y = ex, converts it into which is equivalent to a?^ + ^/^ = 1. 8. Let us now enquire what determination of c as a func- tion of X will convert the primitive <^ {x, y, c) = into a relation between x and y still satisfying the differential equa- tion. Now the complete primitive of a differential equation of the first order is alwa,ys by solution v^^ith respect to y reduci- ble either to a single equation or to a series of equations of the form y=f{^><^) • (!)• If we differentiate, regarding c as constant, we have as the derived equation dy ^ df {x, c) , dx dx ^^^^ and the elimination of c from this by means of the previous dii equation gives us a value of -~ which satisfies the differential equation. That differential equation would then still be satis- fied if c were regarded as variable, provided that the variation were such as to leave unchanged the form of the relation be- AET. 3.] FEOM THE COMPLETE PEIMITIVE. 143 tween x, y and c in the primitive and in tlie derived equation. For the nature of c does not affect the mode of the elimina- tion. Differentiating (1) then on the hypothesis that c is a func- tion of X, and representing the differential coefficient of c thus considered by ( 3~ ) > we have dl ^ df{:x, c) clf(x, c) fdc\ dx dx dc \dxj dv And this will agree in form with the expression for -,- in (2) if ' ' — ^(--J=0. But to suppose j y- J = would be to suppose c a constant and to return to the ordinary primitive. It remains therefore that for a singular solution we have df(x, c) _ di/ „ ,.. This is the first analytical condition. What it means is that if a fixed value be given to x in the primitive, y must not vary for an infinitesimal variation of c. And by this condi- tion c is to be determined as a function of x. Now in accordance with the reasoning of Prop. I. the sub- stitution of a fimction of x for c in a primitive which contains y, cannot lead to a resulting equation not containing y, though it may lead to a resulting equation not containing x. Hence di/ the condition -f^ = can only lead to those singular solutions in the expression of which y at least is involved. Had we reduced the primitive to the form x =f{y, c) we should, as is evident from the principle of symmetry, have arrived at the analytical condition 1 = (5), a condition by which c would be determined as a function of lii DERIVATION OF THE SINaULAR SOLUTION [CH. VIII. y. And the substitution of such value or values of c in the primitive would lead to all singular solutions in the expression of which X at least is involved. It will be remembered that what is essential to a singular solution is that c should not admit of determination as a constant wholly independent of the variables. But whether it be determined as a function of ^r or as a function of y is indifferent. The one form is usually, but not always, con- vertible into the other by means of the primitive. Thus, if the primitive be in the form <^ {x, y, c) = 0, and c be deter- mined in the form c=f{y), the elimination of y between these equations will generally enable us to determine c as a function of X ; but it will not do so if, in the elimination of y, c should disappear. Thus if the primitive were the value of c determined as a function of y by the condition ^^ = would he c = y, and this value of c is not expressible by means of x, for on attempting to eliminate y between the above equations c also disappears. Nor is it indeed possible in the above case to satisfy the condition -^ =0. Hence it is necessary in establishing a general method to take account of both the conditions (4) and (5). And these conditions are sufficient. No other is implied. cl?j The comparison of (2) and (3), from which the condition -f- = uX was derived, leads also to the condition -j- =0, but not to any other condition. The expressions which they furnish for _,- df(x c) become equivalent in two ca^es only, viz. 1st, if ■' ^ ' ^ =0, the case first considered ; 2ndly, if without supposing ART. 4.] FROM THE COMPLETE PRIMITIVE. 145 •^ , = 0, we have • , (i-] infinitesimal dc do \axj dx m com- di (sc Ci parison with • , ' , and therefore if we have dfjx, c) . df(x, c) ^^ ,^. do ' dx for, c being regarded as a function of x, and therefore variable, the factor (-7-) cannot be continuously infinite. Now dif- ferentiating the equation y ^f{x, c) we have dy^^lS^dx + ^I^dc (7). dx do Hence, if we make dy — 0, we have dx _ df {x, c) ^ df{xy c) do dc ' dx (8). SO that (6) assumes the form -y- = 0. But, as a demonstration of this condition, the above method is less general than the previous one, for it assumes the possibility of expressing as a function of x the value of c determined by the condition dx ~ = 0. Now that value is primarily a function of y, and may ac not be expressible at all by means of x. dii dx It is well to note that the final criteria -~ = 0, — = ac etc are in effect analytical expressions of what logicians term con- ditional propositions. The former expresses that if x be assumed constant, y will not vary for an infinitesimal varia- tion of c; the latter, that ifyhe assumed constant, x will not vary for an infinitesimal variation of c, 4. Each of these conditions then dy _ ^ dx _ ^ dc ^ do ' has its special case of failure. The former cannot lead us to B.D.E. 10 146 SINGULAR SOLUTIONS OF [CH. VIII. singnlar solutions in which y is not involved ; the latter can- not lead to those in which x is not involved. It is proper to shew that except in such cases of failure they are equivalent. As expressed by means of the primitive y — f{x, c), these conditions assume the forms df{x, c) ^ Q dfjoo, c) _^ df{x,c) ^ ^ dc * do ' dx do * dc ' dx dif and these are equivalent unless -j- be or infinite. Bnt -^ = implies that the singular solution is of the dx form 2/ = a definite constant, and this is precisely that form of singular solution which the djO condition ;t- = fails to give. Similarly -^^ = 00, being equivalent to -1- = 0, implies that the singular solution is of the form cc = a definite constant, and this is that form of singular solution which the condition -/ = fails to give. do niJ dx Thus the conditions ;/ = ^^ ;/" = ^; although not necessarily equivalent, do not lead to conflicting results. "When we cannot solve the primitive equation with respect to y and x so as to enable us to form directly the expressions AET. 4.] DIFFEKENTIAL EQUATIONS OF THE FIRST ORDER. 147 dii dx for -J- and -7- , we may proceed thus. Representing the pri- mitive by ^ = 0, we have on differeiftiation -j^ dx + -Y^du + -— dc — 0, dx dy ^ do * du d or Hence, remembering what is meant by ~ and -j- , CLC etc (9), d(f) d(j) dy do dx do do ~d4>' dG~ d(j) dy dx and the second members of these equations must be equated to 0. We see that these second members will usually vanish if -^ = 0. And this equation -^ = is adopted by some writers as a sufficient expression of the rule for the derivation of the singular solution from the complete primitive, unrestricted by any accompanying condition. (Lagrange, Galcul des Fonc- tions, p. 207.) We must notice however that the vanishing dxi dor of ~ or ~ in (9) may be due not to the vanishing of the numerator — — , but to the assumption of an infinite value by CtG the denominator -7- or -7^. The latter is indeed quite as dy dx ^ probable a cause as the former when is not expressed as a rational and integral function of x and y. And even when (f> is thus expressed the condition -y^ = may fail through its involving a factor contained in -^ or -7- . We conclude that dif while the true tests of a sinc^ular solution are -f^ = and ° do -T~ = 0, any subsidiary conditions such as 3- = 0, -~ = 00 ^ 10—2 148 SINGULAR SOLUTIONS OF [CH. VIIT. -^ = 00 , are only to be used for purposes of convenience, and never without reference^o the more fundamental relations of which they take the place. The following is a legitimate example of the application of ■ the subsidiary condition -^ — 0. The complete primitive of the differential equation -^ = 2yh 1^ y — {x — cf. Here ^ = y — (x — cY, and, this being rational and integral, the condition -r^ ~ ^ gives 2 {x — c) =0, whence c=x, a value of which, substituted in the primitive, gives ^^=0 a singular solution. clt/ The condition --f- = also gives c=x, and leads to the same result. But, since the primitive solved with respect to iz; gives dbX x = c + yi, the condition -r- =0 cannot be satisfied. Thus the singular solution is here obtained by means of the condition -^ = 0, and not by the condition 3- = 0. 5. The chief results of the above investigation are com- bined in the following Proposition. Peop. II. Every singular solution of a differential equa- tion of the first order may he deduced from its complete prim.i- tive by giving therein to c a variable value determined from that primitive by either or both of the equations dy Q ^ . . And any solution which is thus obtained, and which carinot be also obtained by giving to c in the primitive a constant value, is a singidar solution. The conditions (1) are equivalent, excepit when one only of the variables x and y is involved in the singular solution; solu- ART. 5.] DIFFERENTIAL EQUATIONS OF THE FIRST ORDER. 149 tions involving only the variable y resulting only from the du condition -j- — 0, and those involving only the variable x re- sulting only from the condition -7- = 0. When the primitive, represented by = 0, is rational and integral we may for convenience employ the single condition -j- = 0; but never without reference to the fundamental con- ditions (1). In the statement of the above theorem the two following particulars should be noticed. 1st. It supposes c to be determined as a variable quantity. Now if c be obtained as a function of both x and 7/, as it generally Avill be if the condition -^ = be made use of, it may be necessary by a subsequent elimination to reduce it to a function of one of the variables, in order to assure ourselves that it is not constant in virtue of the relation between x and y established in the primitive. 2ndly. The theorem takes account equally of the positive and of the negative characters of a singular solution. The existence of a variable value of c determined by either of the conditions (1) does not assure us that the resulting solution is singular, unless constant values of c are at the same time excluded. Ex. 1. The equation y^ — 2xy -^ + (1+ x^) (-^j = 1, has for its complete primitive y = cx + ^/{l — c^). Its singular solution is required. Here -^ = x jy:: ^ . Hence -^ = gives for c the do V(l-c) do ^ X variable value c = —rrr, — tt > the substitution of which in the V(^ +1) primitive gives y = ^(x''+l) (1). 150 SINGULAR SOLUTIONS OF [CH. VIIL This value of y satisfies the given differential equation, and it is evident on inspection that it is not included in the com- plete primitive. Formally to establish this, we find on elimi- nating y between that equation and (1) c^ + \/(l-c')=V(^' + l); solving which with respect to c, we have the unique value c = — ^^ — — ^ , which, ag^reeino^ with the value of c before employed, shews that c admits of no other value, and in particular that it admits of no constant value. The solution is therefore singular. doc The condition -7- = would, in the above example, give c — '- ^ , and lead to the same final result. 2/ We must be careful not to rely upon the condition -^ = 0, except under the circumstances specified in the general theorem. This remark will be illustrated in the following example. Ex. 2. The complete primitive of the differential equa- tion y =px -1 — , where p stands for -,'^, is y — ex = 0, and, if we represent its first member by ^, the elimination of c between the equations ^ = 0, -^ = 0, gives the singular solu- tion y^ = 4:mx. But, though this is not a procedure likely to be adopted, if we reduce the primitive by solution to the form ART. 5.] DIFFERENTIAL EQUATIONS OF THE FIRST ORDER. 151 and then represent its first member hy , we shall have dy _ d(l) dcf) dc dc ' dy = 2- ± + 1 I 1 [x x\J {y^ — 4tmx)y And here the singular solution y^ — hmx = 0, before obtained, is seen to be dependent, not upon the vanishing of -~ , but upon the assumption of an infinite value by ~ . «/ The true ground of preference for the conditions -y^ = 0, dc dx -,- = 0, consists, however, not in the directness of their appli- cation to irrational forms of the primitive, but in the plainness of their geometrical interpretation, and still more in their fun- damental relation to the problem of the derivation of the singular solution from the differential equation — questions hereafter to be discussed. The following example is intended to illustrate that portion of the theorem which relates to the negative character of a singular solution. Ex. 3. The complete primitive of the differential equation is y = c(x— c)^ The singular solution is required. dv Here the condition -/ = gives {x — c) {x — 3c) = 0, 152 SINGULAR SOLUTIONS. [CH. VIIL whence c^x, or - . These values of c, both of which are variable, reduce the primitive to the forms 2/ = 0, 2^ = 27 ' and both these are solutions of the differential equation. But while the latter of the two is not included in the complete primitive, the former is included in it. If between the equa- tions y = c[x-c)\ y = 0, we eliminate j/, the resulting values of c will be c = 0, c=x. We see therefore that the solution to which we were led by the assumption c = a? is a particular integral. But it pos- sesses the geometrical properties of a singular solution ex- plained in the following Article. Geometrical Interpretation, 6. Let y—f{pc, c) represent a family of curves the indi- vidual members of which are determined by giving different values to c. Then, adopting for a moment the language of infinitesimals, the differentiation of y with respect to c implies the transition from an ordinate 2/ of one curve to an ordinate y j^ ^^Q^ corresponding to the same value of x, but belonging to another curve of the series; viz. the curve obtained by changing c into c + dc. When we impose the condition -i=^ = 0, we demand that this transition shall not affect the value of the ordinate y corre- dy sponding to a value of x determined by the equation -i^ — ^- AET. 7.] GEOMETEICAL INTERPRETATION. 153 Hence the singular equation obtained by the elimination of c between the equations y — f{x, c), ^'- = 0, represents the locus of such points of successive intersection. In stricter language, the singular solution represents the locus of those points which constitute the limits of position of the points of actual intersection of the different members of the family of curves represented by the equation y =f{x, c), always excepting the case in which that locus coincides with a particular curve of the system. And as at these limitino: points the value of -r- is the same ^ ^ dx for the locus of the singular solution and the loci of primitives, it follows that the former has contact with every curve of the latter system which it meets. The locus of the singular solu- tion is seen to be the envelope of the loci of primitives. The envelope of the loci of primitives is the locus of a singular solution, except when it coincides with one of the particular loci, of which it forms the connecting bond. Similar observations may be made with reference to the condition -y- = 0. do Derivation of the singular solution from the differential equation. 7. We have found that the singular solution of a differen- tial equation considered as derived from its complete primitive possesses the following characters. 1st. It satisfies one of the conditions -^ = 0, -r = 0. do dc 2nd. It is not possible to deduce it from the complete primitive by giving to c a constant value. It has also been shewn that the positive conditions are equivalent except when the singular solution involves only one of the variables in its expression. 154* DERIVATION OF THE SINGULAR SOLUTION [CH. VIII, Now we shall endeavour to translate the above characters from a language whose elements are x,y, and c to a language whose elements are x, y, and -j- , — from the language of the (XiSC complete primitive to the language of the differential equation. If we differentiate with respect to x the complete primitive expressed in the form y^f(x,c).., (1), we obtain the derived equation i' = '^ (2), and substituting in this for c its expression in terms of x and y given by the primitive (1), we have finally the differential eqibation in the form jp = j>{x,y) (3). Thus the differential equation (3) is the same as the derived equation (2), provided that c be considered therein as a func- tion of X and y determined by (1). Accordingly we have |in(3) = fin(.)x|iMl). dy ^ dxdc ' do ' . .^. dc , dy ^ ^ df{x, c) snice m (1) _ = 1 ^ ^- = 1 -r- , . Uj U U/\j too Hence c^ '° (3) = ^ log -^^ , ART. 7.] FROM THE DIFFERENTIAL EQUATION. 155 provided that the value of the first member be derived from the differential equation, that of the second member from the complete primitive. In like m-aiwaar if we suppose the complete primitive ex- pressed in the form we shall have through symmetry the relation d fV\ _ d ^ dx dx\pj dy ^ dc (5), the first member referring to the differential equation, the second to the complete primitive. The equations (4) and (5), which are rigorous and funda- mental, establish a connexion between the differential equa- tion and the complete primitive, and it now only remains to di/ doc introduce the conditions -# = 0, -7- = 0. "We bearin with the do do ^ former. We have seen that when -v^ = leads to a singular solu- dc ^ tion it does so by enabling us to determine c as a function of X, suppose c = X. Before proceeding to more general con- siderations it will be instructive to make a particular hypo- thesis as to the /orm of the equation -y =0. Suppose then this equation to be of the form Q(c-Xr = (6), m being a positive constant and Q a function of a^ and c, which neither vanishes nor becomes infinite when c = X. This hypo- thesis is at least sufficiently general to include all the cases in dy which -^ = is algebraic. 156 DERIVATION OF THE SINGULAR SOLUTION [CH. VIIL By (6) we have then dQ dp d , dy dx v^.^ .^. dy^Tx^'sto^-q-'^T^rx ^^^' and the second term of the right-hand member having c — X for its denominator and not containing c at all in its nume- rator, is infinite. At the same time, we see that no snch infinite term would present itself were c determined as a constant. For let ^^=Q(c- aT, then A log ^ = ^ - Q, the right- do ^^ ' dx ^ dc dx ^ "^ hand member of (7) being now reduced to its first term. The conclusion to which this points is that ^ is infinite for a singular solution, but finite for a particular integral. Again, suppose the value of c in terms of x and y fur- nished by algebraic solution of the complete primitive to be c = Z=0. Now the same substitution gives to the infinite term in the value of --, - the form dy dX — m-j— 'P{^,y)-^ ^ '' We see then, in the case of a singular solution correspond- ing- to a determination c = X, that -/- as derived from the differential equation becomes infinite owing to ^ (a?, y) — X occurring in a denominator. And, whatever modification of form may be made by clearing of fractions or radicals, we may still infer that, if zt = be a singular solution derived from an ART. 7.] FROM THE DIFFERENTIAL EQUATION. 157 alo^ebraic primitive, the function 4- will become infinite, owing to u presenting itself under a negative index. The analysis does not however warrant the conclusion that any relation between x and y which makes -^ infinite will be a solution. If m be a negative constant, the second term in the expression of -J- is still infinite, but the prior condition -^ = is no long-er satisfied. All we can afiirm is that if ~= oo srives a solution at all it will be a sinsjular solution. dy ^ dos 1 Since -y- = - , it is evident that a singular solution origi- ay p ° nating in a determination of c in the form c= Y will make T- (-] infinite. aw \pj dit doc A contrast between the conditions -r^ = 0, ;t- = 0, and the conditions -^ = go , ■!-[-] — ^ > is also developed. The former lead to solutions, but not necessarily to singular solutions ; the latter do not necessarily lead to solutions, but when they do, those solutions are singular. Ex. 1. Given p^ - 2xp + 2y = 0. Here p = x± ^{x^ — 2y), which becomes infinite if 3/ = — , and this satisfies the differ- ential equation. It is therefore a singular solution. It may be objected against the above reasoning, not only that it involves an assumption as to the form of the equa- 158 DERIVATION OF THE SINGULAR SOLUTION [CH. VIII. tion y^ = 0, but also that it takes no accoant of any pos- dc sibilities arising from the first term in the expression of ^ . But it serves well to illustrate what, in the vast ma- dy jority of instances, is the actual mode of transition from the one set of conditions to the other. We proceed to consider the question in a more strict and general manner. 8. "When -^ = determines c as a function of x, it recipro- cally determines ^^ as a function of c, so that if a definite value be given to c, a corresponding definite value or values will be given to x. Let ^ be represented by i/r (x, c), then dp __ d_. dy dy dx do = limit of ^-5S±i^±^.|:^^2g±i^) (9), h approaching to 0. Now for a singular solution '^ (x, c) = 0, and this being, from what precedes, satisfied only by definite values of x, cor- responding to our assumed definite value of c, it follows that '^{x + h, c) will not be equal to for any continuous series of values of h however small ; neither then will log '>Jr{x-^h, c) retain continuously the value of log ^fr (x, c), viz. — oo . Thus the numerator of the fraction in the second member being equal to the difference between a finite and an infinite quantity is infinite, and the limit of the fraction therefore infinite. Hence we conclude that a singular solution considered as derived from the primitive by the conversion of c into a func- tion of X, satisfies relatively to the differential equation the condition dp dy And in the same way it may be shewn that a singular solu- tion derivable from the primitive by the conversion of c into a function oiy satisfies the condition j- f- j = oo ^ ART. 8.] FEOM THE DIFFEEENTIAL EQUATION. 159 Changing the order of the enquiry, let us now examine whether there exist any other forms of solution satisfying the condition -^ = oo,-y-(-) = oo. If there be, it will be made evident that more is involved in the definition of a singular solution than we have yet recognized in our processes of deduction, or else that the definition must be enlarged. Expressing the condition -^ = co , in the form ii°gj = - (10)' we observe that it can be satisfied only in one of two ways, viz. either independently of c, or by some determination of c, and if the latter again only in one of two ways, viz. either by the determination of c as a function of x, or by the determina^ tion of c as a constant. "We may pass over the case in which the above equation is satisfied independently of c, because the relation obtained would involve x only, whereas it has been shewn that -^ = CO leads only to solutions involving ^ at least. We may also pass over the case in which it is satisfied by the assumption c = X, because such a value of c, if it lead to a solution at all, can only do so by satisfying the condition -p = 0, and thus lead to the form of singular solution already investigated. There remains only the case in which the equation (10) is satisfied by a constant value of c. Let then the equation (10) be satisfied hj c = a. The most general assumption we can make respecting the form of its first member is the following, viz. ^logJ = <^(c)f(«,c), where (j) (c) is a function of c which becomes infinite when c 160 DEEIVATION OF THE SINGULAR SOLUTION [CH. VIII. assumes the constant value in question, and ^fr {x, c) does not become infinite for such value. Hence the most general form of loo^ -7^ is ° do ^^^ 'j~— l'^(^)'*k (.^> c)dw= ^ (c) j-^/r (pc, c) dx. To give to this expression the utmost generality, we must, on effecting the integration with respect to x, add an arbitrary - function of c. Thus we shall have log^ = ^ (c) I j^ {oc, c)dx-]rx (c)| . Therefore ^ = g<^(c) {J^(«^,c)cZx+x(c)}, do or, representing the function JT/r (a?, c) dx + x W ^7 ^ (^> ^)> ^ = e*('')*(^'<') ,...(11). dc ' ^ '• This is the most o^eneral form of -^ , as determined from ^ do ' the primitive, which is consistent with the hypothesis that -7- log -^ becomes infinite for a constant value of c, Ac- CLX QiC cordingly if, supposing the primitive to be given, we sought dv to determine the singular solution by the condition t^ = 0, we should be led to an equation of the form gcf- (c) ^ [X, c) ~ Q^ or ^(c)^(^,c) = - 00 (12). Now this equation is not satisfied by any value of c which makes (^ (c) infinite, unless it give to ^ [x, c) an opposite sign * ART. 9.] FEOM THE DIFFERENTIAL EQUATION. 161 to that of <^ (c). But this indicates in general the existence of 'a: relation between x and c. Thus suppose ^ (c) = c, ^ (a?, c) = X. Then (12) becomes C^ = — 00 , which demands that c should receive the value — oo or + oo , according as x is positive or negative. In either case c is constant, but it is a dependent constant — dependent for its sign upon the sign of x. Thus the condition -J- = oo may indicate the existence of a species of singular solution derived from the complete primitive by regarding c, not as a conti- nuous function of x, but as a discontinuous constant, the law of its discontinuity being however such as to connect it with the variations of x. Ex.2. Given r)=^-^^. x Here we find |=^i+i°g^)' •• (13), which is infinite if y = 0. And this proves on trial to be a solution of the differential equation, the true value of the indeterminate function in the second member when y=0 being (Todhunter's Diff. Gal. Art. 158). Now the complete primitive is y = e*"". Hence we see that 2/ = is not a particu- lar integral in the strict sense of that term. The value to be assigned to c is not wholly independent of x. We may there- fore regard 2/ = as a singular solution satisfying the condition dp _ dy'"^' 9. We have said that, in general, the equation (12) in- dicates the existence of a relation between x and c. A case of exception however exists. Representing (c) by G, sup- pose ^ {;x, c), expressed in terms of x and G, to be capable of development in descending powers of (7: suppose, too, that B. D. E. 11 162 DERIVATION OF THE SINGULAR SOLUTION. [CH. VIII. the first term of the development is of the form A C, where A is constant andr> — 1. Then as C approaches infinity, (12) tends to assume the form AC'-^'=--oo, indicating that C, and therefore c, possesses more than one value, real or imaginary. Here, then, the condition -^ = oo Avould accompany a solution possessing this singularity, viz. that it corresponds to a multiple value of c, the arbitrary constant in the complete primitive. It is in fact a species of multiple particular integral. Ex. 8. Given p^ — pxy + y^ log y = 0. Here ^^ a^y ± y V(^^ -^Hy) . therefore dp _ x±\/(x^-4^]ogy) _ 1 dy 2 "^VC^'-^logy) '^ ^' and this is made infinite by 3/ = and by cc^— 4 logy = 0, that is by 2^ = 0, y = e. Both these satisfy the differential equation, and the second is obviously a singular solution. To determine the nature of the first let it be observed that the complete primitive is y = e='-'\ and that this reduces to 2^ = 0, irrespectively of the value of x, by the assumptions c = + go and c = — go . Now this is the only case in which two particular integrals agree. We might in any case, by changing in the complete primitive of an equation c into c^, get two values of c for a particular integral, but then it would be for every particular integral. It is only when the property is singular, that the condition -^ = cc is {satisfied. ART. 10.] FROM THE DIFFERENTIAL EQUATION. 163 It is obvious that one negative feature marks all the cases in which a solution involviner y satisfies the condition —— cc . It is, that the solution, while expressed by a single equation, is not connected with the complete primitive by a single and absolutely constant value of c. In the first, or as it might be termed envelope species of singular solutions, c re- ceives an infinite number of different values connected with the values of a? by a law. In the second it receives a finite number of values also connected with the values of a? by a law. In the third species it receives a finite number of values, determinate, but not connected with the values of x. If we observe that all the above cases, while agreeing in the point which has been noted, possess true singularity, we shall be led to the following definition. Definition. A singular solution of a differential equation of the first order is a solution, the connexion of which with the complete primitive does not consist in the giving to c of a single constant value absolutely independent of the value of a?. Criterion of species. 10. It is a question of some interest to determine whether a given singular solution, u = 0, oi a difterential equation, is of the envelope species or not. On the particular hypotheses assumed in Art. 7, it is shewn that singular solutions of the envelope species possess the fol- lowing character, viz. if w = be such a solution, then -.- becomes infinite though containing a term in which u is presented under a negative index. Now inquiries which are scarcely of a sufficiently elemen- tary character to find a place in this work, indicate (with very high probability) that this character is universal and indepen- dent of any particular hypothesis, and that it constitutes a criterion for distinguishing solutions of the envelope species from others. 11—2 164 CRITERION OF SPECIES. [CH. VIII. As an example of an hypothesis different from that of Art. 7, let us suppose dy Q dc log(c-X)' which vanishes when c = X. We find dQ dX d , dy dx dx dx ^^ dc Q ' (c-A)log(c- -X)' The second term in the right-hand member becomes inde- terminate when c =X, but its true value is oo , and it assumes this value in consequence of c — JT presenting itself with a neofative index. We remark that the fraction ^j ^^r is log(c— A) one which vanishes with c — A in whatever manner c— X ap- proaches to 0, — a consideration which is quite of essential importance. Applying the above criterion to some of the previous ex- amples, we see from the form of -^ in Ex. 1, Art. 7, that the singular solution belongs to the envelope species; in (18) Art. 8, it is implied that the solution is not of that species ; in (14) Art. 9 two species are indicated, the solution y = resulting from log y = — cc being not of the envelope species, while the other solution is of that species. 11. The collected results of the above analysis are con- tained in the following theorem. Theorem. The singidar solutions of a differential equation of the first order (Def. Art. 9) consist of all relations which belong to one or both of the following classes^ viz. 1st. Relations involving y, with or without sc, which make -f^ infinite and only infinite, and satisfy the differential equation, dy ART. 11.] EXAMPLES OF SINGULAR SOLUTIONS. 165 2nd. Relations involving x, with or without y, which make -y- (-) infinite and only infinite, and satisfy the differential equation. When a solution as above defined is actually obtained by equating to a factor which appears under a negative index in the expression of-f-or^y-] it may he considered to belong dy dx \pj to the envelope species of singular solutions. In other cases it is deducible from the complete primitive hy regarding c as a con- stant of multiple value, — its particidar values being either 'Ist dependent in some way on the value of x, or 2ndly independent of X, but still such as to render the property a singular one. We may add that there exist cases in which the characters of different species of solutions seem to be blended together. dr) Thus -~- may admit of both a finite and an infinite value, indicating a duplex genesis of the solution from the complete primitive. It may also happen that the assumption of an infinite value by -j- may be attributed, indifferently, either to a negative index or to a logarithm. And then it should be inquired whether or not the solution is of the envelope species, but marked with some peculiarity arising from a breach of continuity in the mode of its derivation from the complete primitive. The following examples are intended to elucidate particular points either of theory or of method. Ex. 1. Given (1 + ^^ g)' - 2^3/ J + jT - 1 = 0. This equation, first discussed in Brooke Taylor's Methodus Incrementorum, is remarkable as having afforded the earliest instance of the actual deduction of a singular solution from a differential equation (Lagrange, Calcul des Fonctions, p. 276). We shall first explain Taylor's procedure, and afterwards apply the above general Theorem. 166 EXAMPLES OF SINGULAR SOLUTIONS. [CH. VIIL Taylor differentiates the equation, and finding {2(l+.^)|-2^^. = 0, resolves this into the two equations (l+,^|_.^ = 0, 3 = (1). The second of these gives y — ox+h, which satisfies the differential equation provided that h = V(l — a^). Thus the complete primitive is y = ax + fs/(l — a^). The first equation of (1) gives, on eliminating -^ by means of the differential equation, and this he terms the singular solution {singularis qitcedam solutio problematis). To apply to this example the general method, we find £cy + \!{x^ — 2/^+ 1) V x'-^l Hence, ^ - ^^rj^ \x + ^^^, j ^, ^ ^^^ . Introducing the condition -^ = co , we should apparently have the equations but of the second of these, as it does not involve y in its expression, no account is to be taken. The first making -^ infinite whether the upper or the lower sign be taken, and satisfying the differential equation, is a singular solution. Again, as also it is derived from the vanishing of a function under a negative index, it belongs to the envelope species. ART. 11.] EXAMPLES OF SINGULAR SOLUTIONS. 167 We may add that it might be found but less readily from the condition -^ ( - J = x . ax \pj The following example is intended to illustrate the use of the latter condition. Ex.2. Given ^ = ^-^ ax Hence, since p = x'"^, the condition ^ = go cannot be ay satisfied. 3ndition -y- I \p The condition ;t- (-) = oo gives ?2^"~* = X and this is satisfied by 5; = if n be less than 1, but is not satisfied by a; = if 7^ be equal to or greater than 1. Now the differential equation is satisfied by a? = 0, whatever positive value we give to n, as may be seen by expressing it dx in the form -j- = ^^ We conclude therefore that cc = is a ay singular solution of the proposed equation if n be positive and less than 1, but a particular integral if n be equal to or greater than 1. We infer too that the solution, when singular, be- longs to the envelope species. In verification, it may be observed that, if n be not equal to 1, the complete primitive is •^ I —n or a; = ((l-«)(y-c)r». Now if n is less than 1, the index in the second member is positive, and we cannot have a; = unless the quantity under 168 EXAJ^IPLES OF SINGULAR SOLUTIONS. [CH. VIII. the index be made equal to 0. But this would give c = y. Hence, a; = is a singular solution. If n be greater than 1, the index in the second member being negative we cannot have ^ = unless the quantity under the index becomes infinite. But this it does if c is infinite. Here then a? = is a particular integral. lin be equal to 1, the complete primitive is and this is reduced to cc = by the assumption c = 0. Here then also a? = is a particular integral. The following example is intended to illustrate a class of problems in which -j- admits of both a finite and an infinite value. Ex. 3. Given p^ — 2xy^p + 4j/^ = 0. Here we find i5=^2/^±V(^V-V) (1)- Therefore dy 2yH ~V(^'-4r)i and this apparently becomes infinite when y=0, and when ^2 __ 4^^ _ Q^ I Q^ for 2^=o» y=iQ' Let us inquire what are the true values of -j- . x' 1st. If 2^ = r-^ , we find, on substitution and reduction, 16 ART. 11.] EXAMPLES OF SINGULAR SOLUTIONS. 169 which becomes infinite whichsoever sign be taken. Hence, ^ = -— is a singular solution ; and, from the mode of its origin, it is of the envelope species. Sf) . ... 2ndly. If 2/ = 0, the value of -^ in (2) becomes infinite if the upper sign be taken, but assumes the ambiguous form - if the lower sign be taken. To determine its true value, we may expand the fraction ^—^ in ascending powers of 3/ . V(^'-V) We thus find dy 22/n V ^ J) which, as before, gives -^ = 00 when, taking the upper sign, we make y = 0, but on taking the lower sign gives ^y ^^\^ J 2 = - + terms containing positive powers of y. •27 2 And this expression, on making y — 0, assumes the value - . These results lead us to infer that the solution 3/ = 0, originates in two distinct ways from the primitive, which is in this case y = c^ [x — cf. It is evident that this is reduced to y = 0, by either of the assumptions c = and c= x. Hence the solution 3/ = is a particular integral. At the same time it is to be noted that this solution pos- sesses all the geometrical properties of a singular solution. The complete primitive represents an infinite system of para- bolas whose axes are parallel to the axis of y, — whose vertices all touch the axis of x, which thus constitutes a branch of their complete envelope, — and of whose parameters each is inversely as the square of the distance of the corresponding 170 EXAMPLES OF SINGULAE SOLUTIONS. [CH. VIII. vertex from the origin of co-ordiDates. The nearer any par- ticular vertex is to the origin, the more does the curve to which it belongs approach to a straight line; and the curve, if we may continue thus to speak, whose vertex is at the origin coincides with the axis of x which is the envelope of the series. It might in a certain real sense be said that the particular and the general are here united. The following example shews, though by no means in the most extreme case, how slight may be the difference between a singular solution and a particular integral. Ex.4. Given ^ -^ = ?/ (logo; + log y — l). Bepresenting -^ by p, we have y (logo; + logy -1) ^ ■^ X therefore f jogx^lo gy ay X and this becomes infinite, 1st, if y = 0, 2ndly, if y =■ oo , 3rdly, if a; = 0. The first only of these satisfies the differential equation, the assumption y = reducing the indeterminate function y log y in the second member to (Todhunter's Differential Calculus, Art. 158). We conclude, that y=0 is a singular solution, but from the nature of its origin not of the envelope species. Now the complete primitive is y = — , and, judging from X this, it might at first sight seem as if y = were a particular integral corresponding to c = — oo . We remark however that the primitive is not reduced to y = 0, by the assumption c = — 00 , unless x he positive. If x is negative we must make c = + 00 to effect that reduction. In fact, the value of c which reduces the complete primitive to the form y = 0, though in- dependent of X in all other respects, is dependent upon x for ART. 11.] EXAMPLES OF SINGULAR SOLUTIONS. 171 its sign, which must always be opposite to the sign of x. And this connexion, slight as it is, determines the character of the solution. The following example illustrates a mode of procedure which may be adopted when -— presents itself in the am- biguous form - , while the differential equation cannot readily be solved with respect to^. Ex. 5. Given 'p^ — A (p). The singular solution may then be found either as in Chap. VII., or by the direct application of the condition dp _ dy~ ' Geometrical problems which are of a truly symmetrical character frequently admit of this kind of generalization. HemarTcs on the foregoing theory. 12. As the theory of the tests of singular solutions which has been developed in this Chapter differs in many material respects from any that have been given before, it is proper to shew in what its peculiarity consists. To this end it will be necessary briefly to sketch the history of this portion of analysis. Leibn itz in 1694, Taylor in 1715 (see Ex. 1, Art. 11), and Clairaut in 1734, had in special problems, and Euler in 1756 had in a distinct memoir entitled Exposition de quelques Para- doxes du Calcul Integral, examined, more or less deeply, various questions connected with the singular solutions of differential equations. Taylor in particular had first recog- nised the distinctive character of such solutions as set forth in their definition. The problem of the deduction of the singular solution from the differential equation seems however to have been first considered in its general form by La place. The same problem was subsequently investigated in a different manner by Lagr ange, and again in a still different way by Cau chy. The state of the theory up to the present time will be adequately represented by a summary of the results to Avhich these several investigations have led. 1st. Laplace {Memoir es de VAcademie des Sciences, 1772), employing the method of expansions, arrived at results which agree, so far as they go, with those of this Chapter. They ART. 12.] ' OF SINGULAR SOLUTIONS. 175 apply only to the envelope species of solutions, and the demon- strations of them rest essentially on the hypothesis expressed in (6), Art. 7. Lagrange, with whom originated a more fundamental idea of the method of the inquiry, was led to the less exact criteria dp _ dp _ dy ' dx ' ( Calcul des Fonctions, Le9ons xiv — xvii.) Cauchy, whose method was founded on the study of the cases of failure of certain processes for obtaining the complete primitive in the form of a series, was led to the conclusion that a singular solution must satisfy one of the two following conditions, viz. dp _0 dp _ together with a certain further condition, the application of which depends upon a process of integration (Moigno, Calcul, Vol. IL p. 435). Upon these results the following observations may be made, 1st. Although Laplace recognised the necessity of employ- ing in certain cases the condition -y- (-) = oo, for-^ = oo, ax \p/ ay subsequent writers who have employed his method seem to have invariably omitted this qualification. 2ndly. The supposed criterion -^=00, introduced by La- grange, and since very generally adopted, as the proper accom- dp paniment of -^^ = co , is erroneous. If we should apply it to Ex. 2, Art. 11, viz. p = ^"", we should be led to the conclusion that £c = is a singular solution whenever n is positive. We have seen however, both from the application of the true test, and by verification from the complete primitive, that a; = is a singular solution only when n is less than 1. 176 HISTOEICAL ACCOUNT [CH. VIII. The principle of Lagrange's method was the same as that adopted in the present Chapter, and consisted in expressing ~- and ~~- as derived from the differential equation, by means of ay differential coefficients derived from the complete primitive before the elimination of c. The fallacy which vitiated his results consisted in assuming that these expressions become infinite in consequence of the appearance of a vanishing factor in their denominators [Calcul des Fonctions, pp. 229, 232).. Moigno, the expositor of Cauchy's views, also quotes La- grange's^ethoS and results as presented by Caraffa, but without involving any essential variation {Calcul, Tom. II. p. 719). Professor De Morgan, in perhaps the latest publi- cation on the subject, adopts Lagrange's results, expressing, however, only a qualified confidence in his method [Cam- bridge Philosophical Transactions, Yol. IX. Pt. Ii. " On some points of the Integral Calculus"). And he illustrates these results by geometrical considerations which are sufficient to shew that they contain at least a considerable element of truth. Nor should this be thought surprising. For it is plain that Lagrano^e's condition ^- = oo , and the true condition * *= dx —-[-J = 00, are equivalent, except when the singular solu- ax \tj/ tion makes p assume one of the forms and oo . And such cases do exist. Perhaps the peculiar difficulty of this subject has consisted in the faint and shadowy character of the line by which truth and error are separated. dp 13. Of Cauchy's tests the first, viz. -f ="■?., i^iay certainly be set aside. Whenever — assumes an ambiguous form its dy true value or values must be determined. This is illustrated in some of the foregoing examples. Professor De Morgan's observations on this subject in the memoir above referred to, are deserving of attention. The final criterion, which is peculiar to Cauchy's theory, seems to be founded upon what we cannot but regard as an unauthorized position as to the meaning of ART. 14] OF SINGULAR SOLUTIONS. 177 a singular solution. Thus y = 0, the solution deduced by the criterion -^- = oo from the differential equation j)—y log y, is regarded by Cauchy as a particular integral. Now although when X is real the complete primitive log y — c^ reduces to y =^ by the assumption c = - oo , it does not necessarily do so when x is imaginary. Thus, if x^ir sJ{—X), we must make c = ^ , in order to give y = 0. Cauchy's rule seems in- deed to have been designed, contrary to the general spirit of his own writings, to exclude the consideration of imaginary values. Properties of Singular Solutions. 14. Various properties of singular solutions of the en- velope species have been demonstrated. Of these we shall notice the most important. 1st. An exact differential equation does not admit of a singular solution. Let the supposed equation be d4>{x ,y) ^ d^{x,y)dy.^^ dx dy dx ^ '^' and let y =f(x) be a relation actually satisfying it and assumed to be singular. On this assumption the primitive (j) (x, y) = c must, on substituting for y its value f(x), deter- mine c as a function of x and not a constant. Let F(x) be the value of c thus determined, then (f> {x, y) =F {x), whence d4> {^, y) ^ d(f> (x, y) dy ^ dF(x) dx dy dx dx which contradicts (1), since ^7 — cannot be permanently equal to 0, unless F {x) is constant. 2ndly. It follows directly from the above that a singidar solution of a differential equation of the first order ani degree, makes its integrating factors infinite. For let the proposed equation be Mdx + Ndy = 0, ,.„:...:. ..(?!), B. D. E. 12 178 PROPERTIES OF SINGULAR SOLUTIONS. [CH. VIII. and let fx be an integrating factor. Then ti{Mdx-\-Ndy)=-0 (4), TN'ill be an exact differential equation. Hence, a singular solution of (3), while it makes the first member of that equation to vanish, will not make the first member of (4) to vanish. Now comparing these members, this can only be through its making ^ infinite. Ex. The equation ^ + 2/ ;/ = ;/ V(^^ + 2/^ "■ ^) bas for its singular solution x^ -\-y'^ = a^. An integrating factor is and this the singular solution evidently makes infinite. Mul- tiplying the equation by its integrating factor and transposing we have the exact differential equation dy dsG ^y _c\ \l{x'-^-y^ — d') " dx~ ' and this is not satisfied by x^ + ?/" = a^, the singular solution of the unrestricted differential equation. Srdly. Even when we are unable to discover its integrating factor, a differential equation onay he so prepared as to cease to admit of a given singular solution of the envelope species. This proposition is due to JPoisson^ and the following demonstration, which is purposely given in order to illustrate the nature of the assumption usually employed in the theory of singular solutions, does not essentially differ from his. Let us represent the singular solution by w = 0, and trans- form the differential equation by assuming u and x as varia- bles in place of y and x, Suppose the new equation reduced to the form p=f{x,u) (5), where p stands for -j- . '■ ax ART. 14.] PROPERTIES OF SINGULAR SOLUTIONS. 179 This equation is either satisfied or Dot satisfied hj u — 0. If it is not satisfied, the preparation in question has ah'eady been effected. If it is satisfied, the second member y (a?, w) contains some positive power of ic as a factor. Assuming that it can be developed in ascending positive powers of i^ it becomes ^ = Aw" + Bu^ + . . . + &c., where A, B, G, &c. are functions of a?. Now, for a sinofular solution -f- = qo , Hence u = must ° du render Adu"--'^ + B^u^-'^ + &c. = 00 . But this demands that there should exist at least one negative power of u in the above development; therefore a — 1, which is the lowest index, must be negative ; therefore a being already positive must fall between and 1. Hence we are permitted to express the differential equa- tion in the form where a is a positive fraction, and Q does not involve u either as a factor or as a divisor. Dividing by u'^, we have du -, 1 d ^ ^ or — rti-«=Q. l — Oidx Now -z^ = makes v}--°-=0, since 1 — a is positive. Hence the first member of the above equation vanishes, while the second, not containing u as a factor, does not vanish. In its present form then the equation is no longer satisfied by We see also that the property of being satisfied by w = has been lost in consequence of a transformation which, exhibiting the singular solution in the form of a distinct alge- braic factor of the equation, permitted its rejection. See Art. 1. 12-2 180 PROPERTIES OF SINGULAR SOLUTIONS. [CH. VIII. It has been shewn in the remarks on Clairant's equation how, in the process of ascending by differentiation to an equation of a higher order, a somewhat analogous effect is produced, the singular solution seeming to drop aside under charjged conditions. 4thly. Lagrange has noticed that a singular solution will generally make the value of -—^ , as deduced from the differen- tial equation, assume the ambiguous for yn - . His demonstra- tion, in the statement of which we shall endeavour to exhibit distinctly the assumptions which it reall}^ involves, is sub- stantially as follows. Let the differential equation expressed in a rational and integral form be F{x,y,p) = Q (1), then differentiating dp dF . dF Hence -j- = — ^— -^— =ao f.Jj. ay ay dp ^ dF dF Now F being rational and integral, -j- and -r- are so also, and therefore the above can only become infinite for finite dF values of x, y, and p, by supposing -7- = 0. This reduces (2) to the form aF -, dF T rv / A\ -^dx-\- -^dy= (4). ax ay Now, as obtained from the differential equation, d^y _ dp dp dy dx^ dx dy dx dF[ dJ^dy _ dx dy dx " If ' dp ART. 15.] PROPERTIES OF SINGULAR SOLUTIONS. 181 an expression which the previous results reduce to the form - . We may remark that the condition -%- = oo does not involve as a consequence 6^ = go in (2), so as to affect the legitimacy of the deduction of (4). For -~- = oo expresses a conditional proposition, whose antecedent is : li x be constant. Now in the deduction of (4) x is not supposed to be constant. Lagrange's demonstration is certainly only applicable to the envelope species of singular solutions. Of such sohitions it expresses however an interesting property. For the dif- ferential equation being geometrically common both to the locus of the singular solution and to the locus of each parti- d^y cular primitive, the ambiguity of value of —-2 ^^ the point of contact shews that that contact is not generally of the second order. In like manner, i^(^j?/,^) still being supposed rational and integral, the equation dF{sc,7j,p) _ ■ dp ~^ ^^' shews by the theory of equations that the existence of a singular solution implies in general the existence of a series dy of points for which two values of ~ , usually different, come dy . . ... to agree, viz. the values of -f- in any particular primitive, and in the sinsfular solution. 15, Mr De Morgan has made the very interesting remark, that when the condition -/- = go , or -v^ ( in strictness -. — 1 = co , dy dx\ dxpj does not lead to a solution of the differential equation, what it does lead to is the equation of a curve which constitutes the locus of points of infinite curvature (most commonly cusps) ^ 182 EXERCISES. [CH. VIII. in the system of curves represented by the complete primitive (Transactions of the Cambridge Philosophical Society, Yol. ix. Part II.). Geometrical illustrations will be found in the memoir referred to. EXERCISES. 1. The complete primitive of a differential equation is y + c=^ sj{x^ -\-'if —d^), where c is the arbitrary constant. Shew that the singular solution is a? ■\-y^ = a^, and that it may be connected with the primitive by either of the equivalent rela- tions c = — 2/ and c = sjio? — cc^). 2. Why is the above singular solution deduclble by the oljIj ail application of either of the conditions -y- = 0, -^ = ? 3. Expressing the primitive in Ex. 1 in a rational and integral form [x, y, c) = 0, deduce the singular solution by the application of the condition -^ = 0. 4. The complete primitive of a differential equation being x — a = {y — cY, shew that the singular solution is deducible dec by the application of the condition -r —^ but not by that of the condition - '- = 0, and explain the circumstance. 5. The differential equation, whose complete primitive is given in Ex. 1, may be exhibited in the form [x^ — o^) p^ — 2xyp — x^—0. Hence also deduce its singular solution and thereby verify the previous result. 6. Form the differential equation whose complete primitive is given in Ex. 4, and shew that the singular solution is de- d 1 ducible by the application of the condition -^ - = oo but not CH. VIII.] EXERCISES. 183 by that of the condition ^ = x , and explain this circum- stance. 7. Shew that the singular solutions in the last two ex- amples are of the envelope species. Tit 8. The differential equa^tion y = px ■] — (Ex. 2, Art. 5) has y=^cx-\ — ^ for its complete primitive, and ^ = ^mx for its c singular solution. Yerify in this example the fundamental . ,. dp d ^ dy relation ^- = -r loo^ ^ . dy dx ° do 9. Deduce both the singular solution and the complete primitive of the differential equation y =px + ^JQf + a'y), and interpret each, as well as the connexion of the two, geometri- cally. 10. The following differential equations admit of singular solutions of the envelope species. Deduce them. {y — xp) imp — n)— mnp, y={x-l)p-p\ 11. The equation (1 — x^) p + xy — a = Ois, satisfied by the equation y = ax. Is this a singular solution or a particular integral? 12. The equation 2/ = ^ is satisfied by y = 0, which also makes -y- 1 - j = x . Nevertheless v = is a particular inte- dbxypf oral. Shew that this conclusion is in accordance with the general theorem (Art. 11). 13. The equation p (cc^ - 1) = 2a?^ log ?/ has a singular solution which is not of the envelope species. Determine it. 184 EXERCISES. [CH. VIIL 14. Determine also the complete primitive in the lastex- ample^ and shew how the singular solution arises. 15. The equation (p - yY - ^^'y (p-y)= ^^V - 4^y>g y is satisfied by ?/ = 0. Shew that this is a singular solution but not of the envelope species. 16. Find singular solutions of each of the following equa- tions, and determine whether or not they are of the envelope species. 1. jp^ + 2px^ = ^x^y. 2. ocp^ — 2yp + 4a; = 0. 3. xp — n {«** + (?/ — ic") log [y — x"*)]. Geometrical Applications, In solving the following problems, the differential equation being formed, its complete primitive as well as its singular solution is to be found and interpreted. 17. Determine a curve such that the sum of the intercepts made by the tangent on the axes of co-ordinates shall be constant and equal to a. 18. Determine a curve such that the portion of its tangent intercepted between the axes of x and y shall be constant and equal to a. 19. Find a curve always touched by the same diameter of a circle rolling along a straight line. -^ ' 20. Find a curve such that the product of the perpendicu- lars from two fixed points upon a tangent shall be constant. (Euler. See Lagrange, Calc, des Fonctions, p. 282.) CH. VIII.] EXEECISES. 185 . (Representing the product by F, and the distance between the given points by 2m, making the axis of x coincide with the straight Une joining them and taking for the origin of co-ordinates the middle point, the differential equation is {y—{x-^ m) p] \y — {x — m) p\ Its sino-ular solution is o 21. Deduce also the"complete primitive of the above dif- ferential equation. 22. If the primitive of a differential equation be expressed (Til in the form c6 (x, y, a) = 0, the condition y^ = may be ex- pressed in the form ^ ,' ^' ^^ -- r \^' !/> ^) ^ q^ ^^^^ ^ ^ da ay Hence it has sometimes been laid down that ,' ' ' — = go w411 lead to a singular solution. Baabe, in Crelles Journal {Ueber singular e integi^ale, Tom. 48), points out that this rule may fail if at the same time ^ ^ / ^' — - should become in- ' '' da finite. Can it fail in any other case? 23. Exemplify Raabe's observation in the equation a7 + c-V(6c3/-3c')=0, which is the complete primitive of ^xp^ — Qyp -{- x ■]- 2y — . At the same time shew that the singular solutions are y — i^ = and 3?/ + a? = 0. {Crelle, lb.) 24. The complete primitive of a differential equation is (c-x + yy-^{x+y) (c-.T + ?/)'-|-l = 0. 186 EXERCISES. [CH. VIII. Representing its first member, which is rational and integral, by (j>, the condition ;t- = assumes the form S (c - X + y) (c - Sx -y) =0. Shew that e— x + y — O will not lead to a solution of the differential equation at all, while c—Sx — y = will, and explain this circumstance by a reference to Art. 4. Note. The reader is reminded that in all references to the general con- ditions ^ = 00 and -r- (- l = oo, the oo means simply "infinity" irrespec- dy ax \pj tively of sign. See General Theorem, Art. 11, ( 187 ) CHAPTER IX. ON DIFFEEENTIAL EQUATIONS OF AN ORDER HIGHER THAN THE FIRST. 1. The typical form of a differential equation of the n^^ order is given in Chap. I. Art. 2. We may, by solving it algebraically with respect to its highest differential coefficient, present it in the form Its genesis from a complete primitive involving n arbitrary constants has been explained, Chap. I. Art. 8. Conversely, the existence of a differential equation of the above type implies the existence of a primitive involving n arbitrary constants and no more ; and a primitive possessing this character is termed complete. The converse proposition above stated, is one to which various and distinct modes of consideration point, but con- cerning the rigid proof of which opinion has differed. The view which appears the simplest is the following. If, as in Chap. II. Art. 2, we represent by A<^ {x) the increment which the function {x) receives when x receives the fixed incre- ment A^, and if we go on to represent by A^^ (a?) the incre- ment which the function A^ (x) receives when x again receives the same fixed increment A^, and so on, then it is evident that the values of A(/>(£c), A^^(x), &c., are fully determinable if the successive values of the function (j) [x) in its successive states of increase are known. Thus since A0 {x) = {x4- Aa?) — (f> {x), we have by definition A'^ [x) = A{(l>(x+Ax)'-(f> (x)] = [(j)(x + 2Ax) -(j){x + Ax)] - {^(a; + Aa;) -^(x)] = (l>{x+2Ax) - 2<^(a;+ A^) + cfi^x), 188 ON DIFFEREXTIAL EQUATIONS OF [CH. IX. and so on. Conversely if 9(ic), A^(,r), A'(j)(x), &c. are given, the successive values of ttie function ^{x), viz, tlie values (p(x + Ax), (j){x + 2Ax), &c., are thereby made deter- minate. Geometrically we may represent (p {x) by y, the ordi- nate of a curve, or of a series of points in the plane x, y, and therefore functionally connected with the abscissa x. Now the view to which reference has been made is that which, 1st, presents the differential equation (1) as the limit- ing form of the relation expressed by the equation Ax"" ^ V' ^' Ax' A^'-^'^'A X Ax approaching to ; 2ndly, constructs the latter equation in geometry (the arithmetical or purely quantitative construc- tion being therein implied) by a series of points on a plane, of which the first w, viz. those which answer to the co-ordinates oe,x-{- Ax, ... X + (n — 1) Ax, have the corresponding values of y arbitrary, while for all the rest the values of y are deter- mined ; Srdly, represents the solution of the differential equa- tion as the curve which the above series of points in their limiting state tend to form. According to this view, the n arbitrary points in the" constructed solution of the equation of differences (2) give rise to one arbitrary point in the limiting curve, accompanied hj n — 1 arbitrary values for the first n — 1 differential coefficients of its ordinate. And this mode of consideration appears the simplest, because it assumes no more than the definition itself demands of us when we attempt to realize the sreometrical meaning^ of a differential coefficient as a limit. We may however add that when by the consi- deration of the limit, the mere existence of a primitive has been established, other considerations would suffice to shew that in its complete form it will involve n arbitrary coustants and no more. The fact that each integration introduces a single constant is a direct indication of the fact. An indirect proof of a more formal character will be found in a memoir by Professor De Morgan (Transactions of the Cambridge Phi- losophical Society, Vol. ix. Pt. ii.}. , ■ ART. 2.] AX ORDER HIGHER THAN THE FIRST. 189 The above theory may be illustrated by the form in which Taylor's Theorem enables us to present the solution of a differential equation of the ?i*^ order, as will be seen in the following Article. Solution hy development in a series, 2. Reducing the proposed equation to the form dy_r( dy d''^\ dx^^'^V'^'dx'"' d£'-'j"" ^'-^^' and differentiating with respect to x, the first member becomes -, — ^ , while the second member wdll in general involve all ax" the differential coefficients of y up to -y-ft • If ^^^ the last we substitute its value given in (3), the equation will assume the form J""V_// dy d^^ dx"^' "^^ V ^' dx' '"dxH ^^^• Thus ~r-^+i is expressible in the same manner as -~, viz. in (JbJO LLnJU terms of Xy y, and the first n — \ differential coefficients of y. Differentiating (4) and again reducing the second member by means of (3) we have a result of the form d^_.( dy dr2y\ dx"-''~^''\'''^' dx"" dx^-') ^""^^ and in this form and by the same method all succeeding dif- ferential coefficients may be expressed. Hence reasoning as in Chap. Ii. Art. 12, we see that sup- posing y to be developed in a series of ascending powers of x — x^y w^here x^ is an assumed arbitrary value of x, the co- efficients of the higher powers of £c — x^ beginning with {x — x^'*' will have a determinate connexion, established by means of the differential equation, with the coefficients of the inferior powers of x — x^. The latter coefficients, n in number, 190 SOLUTION BY DEVELOPMENT IN A SERIES. [CH. IX. beginning with the constant term which corresponds to the index 0, and endinej with - — -i—-r — — —r-^i , which is the ^ 1.2.S...(n-l) dx''"-' coefficient of {x — cc^Y'^, will be perfectly arbitrary in value. To exhibit the actual form of the development let yo'^/iv Vn-i ^® ^^^ arbitrary values assigned to y, ~ , ... -rn-i when x = Xq, Also lety, /p /a' ^^' i"epresent the values which the second members of the series of equations (3), (4), (5) assume when we make in them x — x^] then y ^yo+ i/A^ - ^o) + Y%i^ - ^oY — + 1.2^(^-1) ^^ ~ ^'^^~' In this expression the arbitrary values of 7/ and its n — 1 first differential coefficients corresponding to an assumed and definite value of ^, viz. y^, y^, -" Vn-i^^^ the n arbitrary con- stants of the solution, the values of/^,/^^^, &c., being deter- minate functions of these, and therefore not involving any arbitrary element. Any function of arbitrary constants is itself an arbitrary constant, and thus it may be that an equation has effectively a smaller number of arbitrary constants than it appears to have from the mere enumeration of its symbols. As a general prin- ciple we may affirm, that the number of effective arbitrary constants in the solution of a differential equation while on the one hand equal to the index of the order of the equation, is on the other hand to be measured by the number of conditions which they enable us to satisfy. Systems of conditions to be thus satisfied will indeed vary in form, but there is one system which we may consider as normal and to which all other systems are in fact reducible. It is that which is de- scribed above, and which demands that to a given value of x a given set of simultaneous values of y and of its differential coefficients up to an order less by 1 than the order of the equation shall correspond. Conversely, the arbitrary constants.; ART. 2.] SOLUTION BY DEVELOPMENT IN A SEEIES. 191 of a solution may be said to be normal, when they actually represent a simultaneous system of values of y and its succes- sive differential coefficients up to the number required, d'li dii Ex. Given -—^ ~ ;7 + 11^' I^equired an expression for y in the form of a series such that when x = 0, y and -—- shall •^ dx assume the respective values of c and c\ Differentiating, we have dx^ dx^ '^'^ dx ^dx'^'^'^^'^^Jx'^^' ^^^ ^^^'^^ equation, = /- + (l + 2^)2. ^^ = 2y| + 2ftV+(l + 2,)g dx^ '^ dx \dxl ' dx = y4-2/+(l + 4,)|+2(|y, by similar reduction, and so on. Hence, corresponding to ir=0, we have the series of values, dy , d^y , ^ g = c^ + (l-f2c)c', ^, = c^ + 2c^ + (l + 4c)c'+2c'^ and so on. Hence, y^c + cx-^r-^ &+d x"^ c''4-(l + 2c)c' 3 . c^ + 2c' + (1+4c)c'h-2c'^ , , . -^—273 ^+— 27374 "'^+^'' 102 FINITELY INTEGRABLE FOEMS. [CH. IX. Finitely Integrahle Forms. 8. As tlie difficulty of tlie finite integration of differential equations increases as their order is more elevated, it becomes important to classify the chief cases in which that difficulty has been overcome. It will be found that for the most part these cases are characterized by some one or more of the following marks, viz. 1st, Linearity, the coefficients being at the same time either constant or subject to some restriction as to form ; 2ndly, Absence of one or more of the variables or their differ- ential coefficients ; Srdl}^, Homogeneity; 4thly, Expressibility in the form of an exact differential or in a form easily re- ducible thereto by means of a multiplier. The subject of linear equations beingof primary importance, we shall devote the remainder of this Chapter to its discussion. But as it will be resumed in another part of this work, and in connexion with a higher method, we propose to notice here only the more important general properties of linear equations, and to illustrate them in the solution of equations with con- stant coefficients. Linear Equations. 4. The type of a linear differential equation of the n^^ order, (Chap. I. Art. 4), is in which the coefficients X, X^ ... X and the second member X are either constant quantities or functions of the indepen- dent variable x. Considering, first, the case in which the second member is 0^ the following important proposition may be established. Prof, li y^,y^,... y^^ represent n distinct values of y, which individually satisfy the linear equation, J + A,^-^. + A,^^.. + A,^ = ....(8), ART. 5.] LINEAR EQUATIONS, 193 then will the complete value of y be Cj', c^, ,..c^ being arbitrary constants. In other words the com- plete value of y is the sum of n distinct particular values of y, each containing an arbitrary constant. For on substitution of the assumed general value of y in (8), we have a result which maybe arranged in the follow- ing form, viz. -^J^■^^S-^^S...-^^, (9). dx"" ' ^ dx""-' ' =^ (^;z;' Now each line in the left-hand member of the above equa- tion is, from the hypothesis as to the values of y^^ y^, ... 3/„, equal to 0. Hence the equation (9) reduces to an identity, and the theorem is established. The problem of the complete solution of a linear equation of the TL^ order whose second member is equal to is, there- fore, reduced to that of finding n distinct particular solutions, each involving an arbitrary constant. 5. Prop. . To solve the linear equation with constant coefficients when the second member is 0. Were the proposed equation of the first order and of the form its solution would be jnx y = c€ B. D. E. 13 194i LINEAR EQUATIONS. [CH. IX. From this result, and from the known constancy of form of the differential coefficients of exponentials, we are led to examine the effect of such a substitution in the equation J + «.^-«-< + c.,^... + «„y = o (10). Assuming then y — Ce"''"^ and observing that ^ye have, on rejection of the common factor Ce*""', the equation m" + a^mT-' + a^nf-^ ... + a„ = (11), the different roots of which determine the different values of m which make y — Ce"*'' a solution of the equation given. When those roots are real and unequal, we have, therefore, on representing them by m^, m^, ... m„, the system of n par- ticular solutions, y = O.e"'', y = Cf-^ ...y= OJ"-" (12), from which by the foregoing theorem we may construct the general solution, 2/=(7/'i^ + C;e'"^^.. + (7„e"'»=^ (13). The equation (11) by which the values of m are determined is usually called the auxiliary equation.^ Ex. Given^,-3^+22/ = 0. Here, assuming y = Ce^'', we obtain as the auxiliary equation m'' - 3w + 2 = 0. Whence the values of m are 1 and 2. The corresponding particular integrals are y = C^e*, and y = C^^''', and the com- plete primitive is ^ ■ ■■ y^C/\C/'\ ART. 6.] I.INEAR EQUATIONS. 195 6. If among the roots, still supposed unequal, imaginary pairs present themselves, the above solution, though formally correct, needs transformation. Let a±h J— 1 represent one of these pairs, then will the second member of (13) contain a corresponding pair of terms of the form which we may reduce as follows, = Ce""" (cos hx + J-lsiB. hx) -f (7'e"'' (cos hx - J~^l sin hx) ^{C+ C) e"^ cos 5a; + ( (7 - (7 V( - 1) e"" sin hx, or, replacing (7+0' and {C — C) ^/{ — l) by new arbitrary constants A and B, ....... Ae''''coshx + Be'"'smhx.. (14). Ex. Given ^-4^+13^ = 0. Assuming y = Ce^^, the auxiliary equation is m'' — 4m + 13 = 0, whence m = 2 ± 8 a/( — 1). The complete solution therefore is T/^^Ae^"" cos Sx + Be^'' sin oX. 7. Lastly, let the auxiliary equation have equal roots whether real or imaginary, e.g. suppose Wg = mj. Then in the general solution (13) the terms (Tjc"^"^ + C^e'^^'^ reduce to a single term ( G^ + C^) e"*!^, and the number of arbitrary con- stants is effectively diminished, since C^ + C^ is only equiva- lent to a single one. Here then the form (13) ceases to be general. To deduce the general solution when m^ — m^ let us begin by supposing m^ to differ from w^ by a finite quantity h, and 13—2 196 LINEAR EQUATIONS. [CH. IX. examine the limit to which the terms of the solution, then really general, approach as h approaches to 0. Now on replacing O^ + C^ and CJi by A and B, new arbitrary con- stants. This change it is permitted to make, however small h may be, provided that it is not equal to 0. The limit to which the last member of the above equation approaches as h approaches to is e'^^'X^ + Bx). This then is the form which must replace (7je*"i^ + C^e'''^'' in the general solution. Suppose next that there exist three equal roots m^, m.^, m^. Then the terms C^e*"^'' + C/^^' + Cge"'^^ being replaced by e''^^ [A ^ Bx) + 0/"^^ make W3 = m^ + h. The above expression becomes = 6"*^'('j.' + ^'^-f (7V+-^a;' + &c.) (15), on making A + c,= a; b+ gjc=e, ^' = c. Here A', B\ C being functions of the arbitrary constants A, B, C provided that k is not actually 0, may themselves be legarded as arbitrary constants. If we so consider them in ART. 8.] LINEAR EQUATIONS. 197 (15) and then make k tend to 0, we see that the limiting form of the expression is And in precisely the same way, were there r roots equal to m^y we should have for the corresponding part of the cortL- plete value of y^ the expression e"^^"" (A^ + A^x + A^x\.. + A^x"^') (16). Thus the difference which the repetition of a particular root m^ produces is that the coefficient of the exponential e"'!"^ is no longer an arbitrary constant, but a polynomial of the form A^ + A^OG + &c., the number of arbitrary constants involved being equal to the number of times that the supposed root presents itself. Ex. Given -7^.- -r.-^+y = 0, ax ax ax ^ Here, assuming y = Ce^"', the auxiliary equation is m^ — m^ — m + 1 = 0, the roots of which are —1, 1, 1. Thus, corresponding to the root — 1, we have in y the term Ce'"", while to the two roots 1 , we have the term (A + Bx) e^. The complete primitive there- fore is y=Ce-''+{A + Bx)e\ 8. It follows from (16), that if a pair of imaginary roots a±h J—1 present itself r times, the corresponding portion of the complete value of y will be ic,+ c,x...+ (7X"') e"^"'^^"' + {g;+ g;x . . . + c;x^') e«^-^^^-\ which, substituting for e^"^"^ and e"*'''^"^ their trigonometrical values and finally making a,+ (7;=-4„ (c,-c;)V^=5„ &c., assumes the form {A^ + A^x . . . + ^X"0 e"" cos hx + (B^ ^B^x,,.-\- B^.x'^') e"^ sin hx. 198 LINEAR EQUATIONS. [CH. IX. Hence, therefore, the repetition of a pair of imaginary roots a±bj—l changes also the two arbitrary constants of the ordinary real solution into polynomials, each of which involves a number of constants equal to the number of times that the imaginary pair presents itself. Ex, Given^,+ 27^^^ + 7?V = 0. Assuming y = Ce*"* the auxiliary equation is m* + 2nV + n* = 0, whence m has two pairs of roots of the form ± n \/(— 1). For one such pair the form of solution would be y — A cos nx-\- B sin nx. For the actual case it therefore is y = {A^-\- A^x) cos nx + {B^ + B^x) sin nx, 9. The above, which is the ordinary method of investi- gating the form of the complete solution when the auxiliary equation involves equal roots, rests on the assumption that a law of continuity connects the form of solution when roots are equal with the form of solution when the roots are unequal. Now, though it is perfectly true that such a law does exist, its assumption without proof of that existence must be regarded as opposed to the requirements of a strict logic. In all legiti- mate applications of the Differential Calculus it is with a limit that we are directly concerned. Here it is with some- thing which exists, and which admits of being determined in- dependently of the notion of a limit. Thus if we take a& an example -y^ "" 2 -~- + y = 0, in which the auxiliary equation m^ — 2m +1 = shews that the values of m are each equal to 1, we are entitled to assume as a par- ticular solution ART. 10.] LINEAR EQUATIONS. 199 Let us now substitute this value of y in tlie given equation regarding G as variable, and inquire whether it admits of any more general determination than it has received above. On substitution we find simply whence C=A + Bx, Thus while the correctness of the solution furnished by the assumption of continuity is esta- blished, it is made manifest that this assumption is not in- dispensable. We shall endeavour to establish upon other grounds the theory of these cases of failure in a future Chapter. Mean- while it maybe desirable to shew that the form (16) actually satisfies the differential equation when r values of m are equal. In, the given equation assume s being an integer less than r. From the theorem for — ^^^^^ it easily follows that the result will be of the form |/(m) x'+f{m) sx'-' -\-f\m) ^ ^^ "l^^"'' + . . . +/^'^ (m) j = 0, in which /(m) represents the first member of (11). But that equation having by hypothesis r equal roots, we know by the theory of equations that * /(m) = 0, / (m) = 0, . . . /'"(to) = 0, are simultaneously true. Thus the differential equation is satisfied. And being satisfied for the particular value of 3/ in (question it is satisfied by (16), which is the sum of all such values. 10. The results of the previous investigation may be summed up in the following rule. 200 LINEAR EQUATIONS. [CK, IX. BuLE. The coefficients being constant and the second mem- her 0, form an auxiliary equation hy assuming y — (7e"**, and determine the values of m. Then the complete value of y will he expressed hy a series of terms characterized as folloios, viz. For each real distinct value of m there will exist a term Ce""*; for each pair of imaginary values a± 6a/(— 1), a term Ae'"' cos hx ■\- Be"'' sin hx] each of the coefficients A, B, C being an arbitrary constant if the corresponding root occur only oncCy hut a polynomial of the (r — 1)*^ degree with arbitrary constant coefficients^ if the root occur r times. Ex. Given?|-^-2^ + 2f^ = 0. ax ax ax ax Here tlie auxiliary equation is m^ — m^ — 2m^ + 2m = 0, Avhence it will be found that the values of m are 0, 1, 1, - 1 + V(- 1). The complete primitive therefore is 2/=C^+(^i+C2^)e*+ C^€~''cosx+ C^e^^'smx. 11. To solve the linear equation with constant coefficients when its second member is not equal to 0. The usual mode of solution is 1st to determine the com- plete value of y on the hjrpothesis that the second member is ; 2ndly, to substitute its expression in the given equation regarding the arbitrary constants as variable parameters ; 3rdly, to determine those parameters so as to satisfy the equation given. Supposing the given equation to be of the w*^ degree, n parameters will be employed. These may evidently be sub- jected to any w — 1 arbitrary conditions. Now that system of conditions which renders the discovery of the remaining re- lation (involved in the condition that the given differential ART. 11.] LINEAR EQUATIONS. 201 equation shall be satisfied) the most easy, is that whicli demands that the formal expression of the n—1 differential coefficients dx' dx'' '" dx""-"- shall, like the formal expression of y, be the same in the sys- tem in which c^,c^, ... c„ represent variable parameters, as in the system in which they represent arbitrary constants. ,The above method is commonly called the method of the variation of parameters. It is, as we shall hereafter see, far from being the easiest mode of solving the class of equations under consideration; but it is interesting as being probably the first general method discovered, and still more so from its containing an application of a principle successfully em- ployed in higher problems. Ex. Given -^ + n^y = cos ax. Were the second member 0, the solution would be y = Cj^ cos nx + c^ sin nx (a). Assume this then to be the form of the solution of the equa- dy tion given, c^ , c^ being variable parameters, but such that -j'- shall also retain the same form as if they were constant, viz. ~ = — c^nsin nx + c^ncosnx (h), Now the unconditional value of -^ derived from (a) is dy . . . de. ^ . dc„ -/- = — c,7i sm nx + cji cos nx + cos nx -~ -^r sm nx -y- , ax ^ dx ax which reduces to the foregoing form if we assume dc dc cosna;-j-^+sinw^-~ = (c). ax ax This then is the condition which must accompany (a). 202 LINEAR EQUATIONS. [CH. IX. Now differentiating (h) and regarding c^, c^ as variable, we have cPy „ 2 • . dc. ^ dc„ —^ = — en cos nx — cji sin nx — n sm nx -r^ + n cos nx -r^ . dx ^ ■ dx dx d^if . Substituting the above values of y and -7-^ in the given equation, we have — n sin wa? -7-^ + 7^ cos nx -^ =^ cos ax (d), dx dx and this equation, in combination with (c), gives <^ti de, 1 . dc„ 1 -t;^ —J = — cos ax sm wic,, -7- = - cos ax cos 7i;r, "^^^ dx n dx n 0^0' , 1 fcos (n + a)x cos (n — a) cc) ^^/4^a^^\'»«mv>'">i Avhence c, = ^- -^^ ^-- + ^ r + ^1 > m * 2n ( n + a n — a J ^ '^J^ _ 1 Jsin (n-}-a)x 8m(n — a)ai\^ ^ ^~ 2n\ 71 + a n — a J ^* Lastly, substituting these values in (a) and reducing, we have cOo ax /~f . /^ ' / \ y= -^ 5+ C, eosnx+ u^smnx (e). ^ n^— a^ ^ ^ This solution fails iin = a. But giving to {e) the form cos ax — cos 7i:c ^ , . >-v • 7/ = + 6V cos nx + G' sm ?ia;, •^ n^ — a'^ ^ and regarding the first term as a vanishing fraction when n = a, we find V = — \- C, cos nx + C' sm Tza?. Or we might proceed thus. Differentiating twice the equation ^^y , 2 —4 + n 1/ = cos nx. dx' ^ d^y 2^y 2 we get ,-4 + 71 -T^ = — n cos wo?. . . _ *=* dx' dx^ AKT. 12.] LINEAR EQUATIONS. 203 Hence eliminating cos nx an equation wliose complete solution is y =^[A-\- Bx) cos wa; + ((7+ Dx) sin nx. Substituting this in the given equation we find jB = 0, I>= —~ . whence zn y = A cos7za;-{-[(7+x-J si sm nx, \ 2w/ which agrees with the previous solution. The latter method, which is general, consists in forming a new equation of a higher order, but with its second member free from that term which is the cause of failure. As by the elevation of the order of the equation superfluous constants are introduced, the relations which connect them must be found by substitution of the result in the given equation. 12. To the class of linear equations with constant coeffi- cients all equations of the form (a+6..)»g+^(a+6#-g|+5(a+6a,r^...+Z2,=X, A, B, ... L being constant and X a function of x, may be reduced. It suffices to change the independent variable by assuming a-\-hx = e\ Ex. Given (a + 6a;)'^ ^ + 6 (a + 5^) || + w^^ = 0. ' Assuming a-\-'bx = e% we find dx dt* 'd = &V X fd^y dy\ \dr " dij ' /// 204 LINEAR EQUATIONS. [CH. IX. Hence, by substitution in tbe given equation, we have the solution of wbicb is n "Ml ^, . tit y—C cos T- + ^ sm j- , in which it only remains to substitute for t its value log (a+6x). 18. Beside the properties upon which the above methods are founded, linear equations possess many others, of which we shall notice the most important. We suppose, as before, y to be the dependent, x the independent variable. 1st. The complete value of y when the linear equation has a second member X will be found by adding to any particular value of y that complementary function which would express its complete value were the second member 0. Representing the linear equation in the form (7), let y^ be the particular value of y which satisfies it, Y the complete value which would satisfy it were the second member ; and assume y =y^-\- Y, The equation then becomes d^ Y cZ""^ Y \=X. (17), and this becomes an identity, the first line of its left-hand member being by hypothesis equal to X, and the second line equal to 0. Ex. Thus a particular integral of the equation ^.-ay^x^l x-\-\ , being y — — • — a~ > i^s complete integral is ^ a ART. 13.] LINEAR EQUATIONS. 205 The above property, wliich relates to the generalizing of a particular solution, is important, because, as we shall hereafter Hee, a particular solution of a linear equation may often be obtained by a symbolical process which does not involve even the labour of an integration. 2ndly. The order of a linear differential equation may always be depressed by unity if we know a particular value of y which would satisfy the equation were its second member equal to 0. It will suffice to demonstrate this property for the equation of the second order g + X.| + X., = Z (18). Let 2/j be a particular value of y when X= 0, and assume y ~y^P' Substituting, we have the first line of which is by hypothesis 0. In the reduced equation let -7- = ^> then we have ax Vx £+(4'+^'^.)"=^ (1=^)' a linear equation of the first order for determining m. And this being found, we have v-V udx + c. In the particular case in which X= 0, we find from (19) u = y: -jXidx W hence y=y. [cf-^dx+C^ (20). 20G LINEAK EQUATIONS. [CH. IX. Srdly. Linear equations are connected by remarkable ana- logies witli ordinary algebraic equations. This subject bas been investigated chiefly by Libri and Liouville, who have shewn that most of the characteristic properties of algebraic equations have their analogues in linear differential equations. Thus an algebraic equation can be deprived of its 2nd, 3rd,...r*'' term by the solution of an algebraic equation of the 1st, 2nd,...(r — l)**" degree. A linear differential equation can be deprived of its 2nd, 3rd, . . . r*^ term by the solution of another linear differential equation of the 1st, 2nd, . . . (r — l)**" order. This may be proved by assuming y=vy^, and properly de- termining V so as to make in the resulting equation y^ assume the required form. Again, as from two simultaneous algebraic equations, we can by the process for greatest common measure obtain a de- pressed equation satisfied only by their common roots, so from two simultaneous linear differential equations we can by a formally equivalent process deduce a new equation of a de- pressed order satisfied only by their common integrals. This is best illustrated by example. Ex. Required the common integrals, if any, of the equations dx^ dx Differentiating the second equation and then eliminating 73 J^ -f{ and ;t4 j we find the depressed equation clx ctx dx ^ If we differentiate this we shall find that the result is merely an algebraic consequence of the two equations last CH. IX.] EXERCISES. 207 written, not an algebraically new equation. Thus the process of reduction cannot be repeated. We have therefore as the only common integral. [See the Supplementary Volume, Chapter xxii.] EXERCISES. ax ax ax ax 3. Integrate^. -4-y4 + 6:T^, -4 :7^4-v = 0. ° ax ax ax ax ^ 5. -y^ — 3 -74 + ^y — 0, it being given that one of the roots of the auxiliary equation, m^ — Sm^ + 4 = 0, is — 1. d^_^dly dly dji ^- dx' ^dx'^^dx^ ^dx^y-^' ^- dc^^^^dx^'^y-'' 8. "What form does the solution of the above equation assume when h^W y. sc "7 s -" X ~^ -— ■ oi/t ax ax ^ 10. (a; + a)^g_4(a. + a)J + 6^/ = 0. 11. Integrate ^-2&a;^ + 6'a:V = 0. 208 EXERCISES. [CH. IX. 12. A particular Integral of (1 — a;^) -^^ —x-~— a^y — is y= (7g«s«'^~*^^ find the complete integral by the method of Art. 13. . 13. The form of the general integral might in the above case be inferred from that of the particular one without em- ploying the method of Art. 13. Prove this. 14. It being given that . ( . cos x\ T^ f sin x\ y=-a ( smxH 1 +ij (cosic J d^y t 2\ is the complete integral of the equation -^^ + f 1 2 ) ^ = ^> find the general integral of vl + ( ^ 2 ) ^ x\ 15. Explain on what grounds it is asserted that the com- plete integral of a differential equation of the rf^ order contains n arbitrary constants and no more. 16. Mention any circumstances under which it may be advantageous to form, from a proposed differential equation, one of a higher order. In deducing from the solution of the latter that of the fornier, what kind of limitation must be introduced? ( 209 ) CHAPTER X. EQUATIONS OF AN ORDEE HIGHER THAN THE FIRST, CONTINUED. 1. We have next to consider certain forms of non-linear equations. Of the following principle frequent use will be made, viz. // When either of the primitive variables is luanting, the order of the equation may he depressed hy assuming as a dependent variable the loivest differential coefficient which presents itself in the equation. Thus if the equation be of the form 4'2'S)=« W' F {x, z, and we assume i- (2).. we have, on substitution, the differential equation of the first order, S)-« -(3)- If, by the integration of this equation, z can be determined as a function of x involving an arbitrary constant c, [suppose z = (j)(x, c)}, we have from (2) 2,= (x, c), whence integrating y = j (p {oc, c) dx + c, B.D.E. 14 210 EQUATIONS OF AN OEDER HIGHER [CH. X. If the lowest differential coefficient of y which presents itself be of the second order, the order of the equation can be depressed by 2, and so on. A similar reduction may be effected when x is wanting. Thus, if in the equation of the second order . ^(^.|'S)=« w- we assume — =p,we have ax cTi!/ __dp _dp dy _ dp dx^ dx dy dx -^ dy' by means of which (4) becomes F{y,P,pfy<> (5). Should we succeed by the integration of this equation of the first order in determining^ as a function of y and c, sup- dif pose p = ^{y, o), the equation -f-=P will give whence X=f;^ + C' (6). Ex. Suppose l + gy+ 2, g = 0. Put , - = p ; thus dx ^ therefore — + :; -^ = 0, therefore log 2/ + log V (1 -Vp") = constant. ART. 2.' THAN THE FIRST, CONTINUED. therefore y^^/(l+ p^) = constant = J, therefore 1+/=,^. therefore dx y . dy~'J{b-'-f)' therefore x = -^{V-f)+a, where a is a constant ; thus finally, f^-{x-af = V. 211 2. In close connexion with the above proposition, stand the three following important cases. Case I. When but one differential coefficient as well as but one of the primitive variables presents itself in the given equation. d^ii 1st. Let the equation be of the form -j-^^ = A^, we have by successive integrations d''-'y _ and finally y=\\ ... Xdx''-\-c^x''-'-Vc^x''-' ... +c„ ......(7). We shall hereafter shew that the first term in the second member may be replaced by a series of n single integrals. 2ndly, If the equation be of the form -~ = Y, it is not generally integrable, but it is so in the case of n = 2. Thus there being given dx'~ ' 14—2 212 EQUATIONS OF AN ORDER HIGHER [CH. X. we have dx dx^ dx * and integrating Hence da!== {2iYdy + Cf J (2 fVdv 4- C^^^ ^ ^' "^ '{2JYdy + C)i d^y As a particular example, let -—^ = <^V Here x = (- ^ ~ + C i{2Sa'ydy+C)^ " = Jlog{a3/+V(ay+a)}+r. ^ Case II. When the given equation merely expresses a relation between two consecutive differential coefficients. Suppose the equation reduced to the form dx''~~J\dx''-') ^ ^' then, assuming ,,_f = ir, we have whence dx = /i^)' =1/1^-^- ^''^- ART. 2.] THAN THE FIRST, CONTINUED. 21 o If, after effecting the integration, we can express z in terms of X and c, suppose z=^{x, c) we have finally to integrate ^-i = ^(^,c) (11), which belongs to Case I. But if, after effecting the integration in (10), we cannot algebraically express z in terms of x and c, we may proceed thus. From ;r^ ~ ^' ^^ have 'zdz =/; dz Czdz - f_ f- /(^)i/W' and finally. f dz f dz r zdz ,- £.. the right-hand member indicating the performance of 7i — 1 successive integrations, each of which introduces an arbitrary constant. If between this equation and (10) we, after integra- tion, eliminate z, we shall obtain a final relation between y, oc, and n arbitrary constants, which will be the integral sought. Ex. Given «Sg = A/{l + (g)}- d^fj dz Making -t^ = z,wq have «^ ;t- = V(l + ^^)> whence ic = c + aV(l +^') W« 214 EQUATIONS OF AN ORDER HIGHER [CH. X. According to the first of the above methods, we should now solve this with respect to z, and thus obtaining find hence si-//y{(^f-i} ^c* That equation will therefore have its order depressed by unity on assuming J=p...(Art.l). Let us examine the general form of the result for equations of the second order. Eepresenting the given equations under the form ^(-^-2'S)=« • (1^)' we have, on substitution, and from this equation, from what has been above said, e^ will disappear on division by some power of that quantity, e.g. e''^. But the effect of simply removing a factor is the same as that of simply replacing such factor by unity. Now to replace e"^ by ART. 3.] HOMOGENEOUS EQUATIONS. 217 unity is the same as to replace e^ by tmity, and if we do this dz d z simply, i.e. without changing -7^ and -^ , (17) will become ^^-S-^^'S+S)=« (i«)- dz d^z dz' Id'^^'W^dd. dz , d^z du du Assuminor then -j7. = u, whence -y^o = "tti = ^ ^- > we have ° du dd do dz F(l, z, u + z, u^ + u)==0 (19), an equation of the first order, which by integration gives u = cj,{z,c) (20). dz Then since ^l = -T7^^ we have CtC7 l>g)=^ • ^''^' the reduced equation for determining u will be Suppose that by the solution of this we find u = {z,c)„... (25), then since ^ ~ ^ ' ^^ '^b.yq de ^' (^, c) ' = [;j-J^ + c' (26), («. c) in which itonly remains to substitute log x for 6, and ^ for z, sty Ex. Given x'^, = {x''{-2xy)^-iy\ This equation proves homogeneous on assuming x to be of the degree 1, y of the degree 2, -^ of the degree 1, and (I'll -— of the degree 0. Changing then, according to the" formula (24), x into 1, y into z, -f^ into w + 2z, and -r^ into ^^ ^- + S^t + 2^, we have •^ dx dx dz u^ + Su + 2z= [1 + 2z) {u + 2z) -- 4iz' (a), 220 HOMOGENEOUS EQUATIONS. [CH. X. wliicli is reducible to .g-H2-2.)=0. This is resolvable into two equations, viz. dz The first gives on integration u^[z-Vf±c\ Hence, since -Tn = '^} ^^ have nil ;^ + 2-2^ = 0, u = Q (6). {z-rf±c^' Hence, replacing 6 by log a?, and ^ by — 2 , we have log oj = - tan"' ^^^ — 2- + c', or - log ^ — )- f^ + c', ° c cic^ 2c ^ ^ — (1 — c) a;^ ' the rational forms of the integral required, dz d9 dz The factor w = in (6) giving ^ = 0, or ^ = c, leads to the singular solution y = ca?, ^ Class III. Equations which are homogeneous with re- spect to ^,^, ^,&c. Properly speaking, this class constitutes a limit to the class just considered. For when n becomes large, the quantities 71, n — 1, ?^ — 2, the supposed measures of the degrees of ?/, -7- , v| approach a ratio of equality. ART. 8.] HOMOGENEOUS EQUATIONS, 221 If we assume y — e", we have dy , dz e dx dx (2V), 3-'{S+{l)"} » All these being of the first degree with respect to e", it fol- lows that after substitution in the proposed equation, that function will disappear on division. Thus, if the given equa- tion be ^^'^'2'S)=« (^9). the transformed equation will be dz d'^z dx ' dx^ ' \dx, ■n 1 ^ dz d'^z fdzV] - ,„^, or, on assummof -r- = w, ' ^ dx ^ dz du fU 1, u, ^ + ^') = (81). Integrating this equation of the first degree, we have u = ^(Xf c) ; therefore z = I cf) (x, c) dx + c (32), in which it only remains to substitute for ^ its value log y. Or we may assume at once y = e'' . The transformed equation between u and x will be of an order lower by unity than the equation given. dy d^V fdv\^ dx Ex. Given ay-j^.-^hi-^] — , , ., rr . ^ dx' \dxj V (e + x') Assummg y = e , we find ^du , „\ u , 2 \dx J V(6 +a;j 222 EXAqT DIFFERENTIAL EQUATIONS. [CH. X. as would directly result from (29) and (30). Expressed in the form = -! + -< dx a^/{e^ + x^) \ a this equation is seen to belong to the class discussed in Chap. II. Art. 11. \ On comparing the above classes of homogeneous equations we see that Class II. is the most general. It includes Class I. as a subordinate species, and Class III. as a limit. It is proper to observe that Classes I. and ii. are usually treated by a different method from that above employed. Thus, in Class i., it is customary to make the assumptions _ dy _ d^y _ v d^y _w ^ On substitution x divides out, and there remains an equa- tion involving y and the new variables t, u, v, w, &c,, which may be reduced by successive eliminations to a differential ' equation between two variables, and of an order lower by ■unity than the equation given. But this method is far more complicated than the one which we have preferred to employ. Exact Differential Equations, 4. A differential equation of the form is said to be exact if, representing its first member by V, the expression Vdx is the exact differential of a function tf, which is therefore necessarily of the form '^ i^, y, -f- , .-• 'T^^ij • Thus ~ —K — vx^ ~ — xy^ = is an exact differential dx dx ^ dx ^ equation, its first member multiplied by dx being the differ- ential of the function ^\\--f) — ^^li\ > ^^^ ^^^ ^^^^ member itself the differential coefficient of that function. AET. 4.1 EXACT DIFFERENTIAL EQUATIONS. 223 Hence then a first integral of the above equation will be The rhethod of integrating an exact differential equation which we shall illustrate, and which contains an implicit solution of the question whether a proposed equation is exact or not, appears to be primarily due to M. Sarrus {Lioimlle, Tom. XIV. p. 131, note). Ex. Given ,.3.J^2,(i)V(.^+v|)g = 0. Supposing the above an exact differential, we are by defi- nition permitted to write Now a first and obvious condition is that the hig-hest differ- ential coefficient m an exact differential equation, being the one introduced by differentiation, can only present itself in the first degree. This condition is seen to be satisfied. Kepresenting the highest differential coefficient but one by p, we can express (34) in the form dU={y-{- Sxp -f 2!//) dx + {x^ + 2fp) dp. Now let ?7j represent what the integral of the term con- taining dp would be were p the only variable. Then Z7j = x^p -f y^p\ Assume, then, removing all restriction. dx "^ \dxj whence Subtracting this from (84) dU-dU, = (2/ + «2 dx (35). 224} EXACT DIFFERENTIAL EQUATIONS. [CH. X. We remark that the highest differential coefficient -r4 has now disappeared. We observe too that the next, viz. -^ is involved only in the first degree. This is a consequence of the fact that the proposed differential equation was really exact. For the first member of (35) being the difference of two exact differentials, and therefore itself exact, the second member is so, and its highest differential coefficient is therefore of the first degree. The integration of an exact differential involving vl has, in fact, been reduced to that of an exact differential ax involving only -^ as its highest differential coefficient. And a similar reduction may be effected whatever may be the order of the highest differential coefficient. ^ The integration of (35) gives whence A first integral of the given equation is, therefore, -s+^iiy+«'^=« • (^6)- The general rule for the integration of an exact differential dU, involving x, y, -^, ... -y-*^-, is then as follows. Integi^ate the term wliich involves -r^ in the first degree, as if -fi^ were the only variable, and -— dx its differential. Representing the result by C^, and removing the restriction, dU—dU^ will he an exact differential involving only x, y, -f '>"- 'Tl^^ * I^&P^cdi the process as often as necessary. Then U will he expressed hy the sum of its successively determined 2'>ortions (J^y U^, U^, <&c. ART. 5.] EXACT DIFFERENTIAL EQUATIONS. 225 For the solution of an exact differential equation^ it is there- fore only needful to equate to c the integral of the correspond- ing exact differential as found by the above process. The failure of that process, through the occurrence of a form in which the highest differential coefficient is not of the first degree, indicates that the proposed function or equa- tion is not ' exact.' 5. There is another mode of proceeding of which it is proper that a brief account should be given. Kepresenting ^, -^,, ... ^, by y,, y,,..-y^, it is easily shewn by the Calculus of Variations, that if Vdx be an exact differential, V being a function of x, y^y^^...y^, then identically dy \dx] dy^ \dx) dy^"^ \dx) dy^ ^ ^' where i-j- J indicates that we differentiate with respect to x regarding y, y^,."y^ as functions of x. This condition was discovered by Euler. The researches of Sarrus and De Morgan, not based upon the employmeat of the Calculus of Variations, have shewn, 1st, that the above condition is not only necessary but suffici- ent. 2ndly, that it constitutes the last of a series of theorems which enable us, when the above condition is satisfied, to reduce Vdx to an exact differential in form, i.e. to express it in the form dU . dU. dU . . dU J ,^^. -dx^-^-^dy^^^-dy,,,,^j^_dy^., (38), where x, y,y^, ».. ^/^.^are regarded as independent. The inte- gration of Vdx = in the form Z7= c is thus reduced to the integration of an exact differential of a function oi n-\-\ inde- pendent variables, — a subject to be discussed in Chapter xii. {Cambridge Transactions, Vol. ix.) The condition (37) is singly equivalent to the system of conditions implied in the process of Sarrus. The proof of this equivalence a 'posteriori would, as Bertrand has observed, be complicated. (Liouville, Tom. xiv.) B. D. E. 15 226 MISCELLANEOUS METHODS AND EXAMPLES. [CH. X. The solution of the differential equations of orders higher than the first is sometimes effected by means of an integrating factor fjL, to discover which we might substitute yu,Ffor Fin (37), and endeavour to solve the resulting partial differential equation. Even here, however, the process of Sarrus would be preferable. Miscellaneous Methods and Examples. 6. Many forms of equations, besides those above noted, can be integrated by special methods, e. g. by transformations, variation of parameters, reduction to exact differentials, &c. Equations of the classes already considered can also sometimes be integrated by processes more convenient than those above explained. d^y Ex. 1. Given -y^ = ax-\-hy. Let ax -\-hi/ = t. We find as the result, -^2 =^i ^ linear equation with constant coefficients. Ex. 2. Given n-x')'Pi-x^ + ohj = 0. Changing the independent variable by assuming sin~^ x = t, we find -7^ + ^y — 0; whence the final solution is 2/ = Cj cos {q sin"^ x) + c^ sin {q sin"^ x) (39). So too the equation (1 + ax^) -r^z-V ax -— ± q^y = 0, is re- d% ducible to the form ^^ + c^y = 0, by the assumpt dx I ^(l+ax') = t Equations involving the arc s, whether explicitly or im- plicitly, may be freed from it by differentiation or by change of independent variable. ART. 7.] MISCELLANEOUS METHODS AND EXAMPLES. 227 Ex. 3. Given s= ax-\- hy. Differentiating, we have a/|i + (^) j = ^ + ^ ^ ' therefore dy ^ ah ± s/{a^^¥ -1) theretore ^^- j _ ^2 , Ex. 4. Given -^^ = a. Assuming x as independent variable, we have d^x _dx d dx __ /ds\~^ d /ds\~^ ds^ ds dx ds \dx) dx [dxj __(ds\~'d^_ \dxj daf ds We might here put for-j- its value aJ(Jl -\-p^), and so form a differential equation for determining p. Direct integration, however, gives -y- ) = 2ax + c. Ax Whence we find dy r" dx \^ax + c which indicates a cycloid. 7. M. Liouville has shewn how to integrate the general equation -y4 +/ (a?) -^ + i^ (3/) ( -7- ) =0 {Journal de Mathema- tiques, 1st Series, Tom. VIL p. 134). Suppressing the last term, the resulting equation dx^ ^ ^ ^ dx 15—2 228 MISCELLANEOUS METHODS AND EXAMPLES. [CH. X. has for a first integral ~- = Ce'^^^^'^'^''. Now assume this to be dx a first integral of the given equation regarding C as an un- known function of y, then dx' \dy dx ^ ^ ^ _ 1 dy (dC dy ^j, , . C dx [dy dx •/ ^ ^ ^'^dG(dyV_ , dy ^ dy \dxj *^ ^ '^ dx' Thus, the given equation becomes 1 dP c'^+^W = o (^0)' wb ence C = Ae S^^y^^y. Therefore ^ = Ae-J'^'^y^^^ x e-/-^^-^''* : dx therefore Lf^(y)ay dy = A je--^^^''^^'' dx + B (41), the complete primitive sought. 8. Jacobi has established that when one of the first inte- d\ grals of a differential equation of the form -t^>i — f{xy y) is known, the complete primitive may be found. The following demonstration of this proposition is due to Liouville {Journal de Mathematiques, 1st Series, Tom. xiv. p. 225). dv Let the given first integral be --^ = ^ (a;, y, c). Differenti- ating, we have d^y _ d(j> d(j) dy _dcj> d(f> dx^ dx dy dx dx dy* if) standing for <^ (a?, y, c). Hence, comparing with the given equation. ART. 9.] SINGULAR INTEGRALS. 229 and differ en tiating with respect to c, dxdc cCc dy dydc Now this is precisely the condition which must be satisfied in order that the expression ~ (dy — ^dx) may be an exact differential. Hence, the first integral expressed in the form dy — dx) = C (42). Some equations of great difficulty connected with the theory of the elliptic functions are reduced to the above case in the memoir referred to. /./ y.^^ Singular Integrals. 0"k^-i^-,^^t ' ^L^-* 9. Equations of the higher orders, like those of the first order, sometimes admit of singular integrals, i.e. of integrals not derivable from the ordinary ones without making one or more of their constants variable. We shall term such integrals singular solutions when they connect only the primitive variables, but singular integrals when they present themselves in the form of differential equations inferior in order to the equation given. And as the entire theory is involved in the theory of singular first integrals, we shall speak chiefly of these, but with less detail than in the corresponding inquiries of Chap. Viii. [Additions to the present Chapter are given in the Sup- plementary Volume, Chapter xxiii.] Prop. Given a first integral with arbitrary constant of a differential equation of the n)'^ order, required the correspond- ing singular integral. Let the given equation be F{x, y, y^, 2^2 ...J/n) = (4:3), 230 SINGULAK INTEGEALS. [CH. X. where y^ stands for —■ , y^ for -^ , &c. Suppose the integral given to be expressed in the form yn-i^f[^> y^ Vx'" 2/.-2' c) (44), c being an arbitrary constant. Differentiating as if c were an unknown function of Xy \^d£ df dl _df_ dfdc^ ^« dx ^ dy ^^ ^ dy^ y^'" "^ dy^J^"'^ dc dx ' Now this reduces to the same form, i.e. gives the same expression for y^ in terms of x, y^ ••• 3/n-i5 c, ^s it would do if c were constant, provided that we have -j = 0; and therefore, this condition satisfied, the elimination of c will still lead to the given differential equation (43). An integral of the given equation will therefore be found by attributing to c in the complete first integral (44), such df value as will satisfy the condition -^ = 0, or, as we may ex- press it, %•=« "(^s)- And unless the value of c thus found is constant, the integral will be singular. The above process amounts to eliminating c between (44) and (45), so that we have the following rule. Given a first integral of a differential equation of the ^* order, to deduce the corresponding singular integral, we must eliminate c between the first integral in question and the equa- tion -4""^ = ^> where y^__^ is the value of , ^_^ expressed in terms of x, y ... -.^^ , &c. hy means of the given first integral. If the proposed first integral is rational and integral in AET. 9.] SINGULAR INTEGRALS. 231 form, then representing it by ^ = 0, it suffices to eliminate c between the equations, * = 0,f = (46). It is unnecessary to dwell on the particular cases of excep- tion after what has been said on this subject in Chap. Vlii. Ex. 1. The differential equation y-^yi+ ^y-z- iyx - ^yS - y^ = ^> has for a first integral required the corresponding singular integral. Differentiating the first integral with respect to 6, we find whence h=-~~ ^ , and this value substituted in the p^iven 4 (1 + x') ' ^ integral, leads to ,, ^^1 y" , {^fx-^'f ^0 y 2 x''^ IQx' (1 + x') ' . or, on reduction, 16 (1 + x^y - 8xSj^ - 16x1/^ +x'- 16?/,' = 0. In connexion with this subject, Lagrange has established the following propositions : 1st. Either of the first two integrals of a differential equa- tion of the second order leads to the same siDgular integral of that equation. 2nd. The complete primitive of a singular integral of a differential equation of the second order will itself be a sin- gular solution of that equation, but a singular solution of a singular integral wil] in general not be a solution at all of that equation. 232 SINGULAK INTEGRALS. [CH. X. The proof of these propositions will afford an exercise for the student. [See Lagrange's Legons sur le Calciil des Fonctions, Legon 14"^' of the edition of 1806, or Legon IS""" of the edition of 1808. A note by Poisson on page 239 of the edition of 1808 should be consulted ; it relates to the second of the above two propositions. See also Lacroix, Traite du Calcul Diffe- rentiel ..., Tome ii. pp. 382 and 390.] 10. We proceed to inquire how singular integrals may be determined from the differential equation. Expressing as before the first integral involving an arbi- trary constant in the form we have as the derived equation ^^^j ^/(.,,,y.^..V...,c) | ^^g^^ the brackets in the second member indicating that in effect- ing the differentiation y, y^, '"Vn-^v ^^® ^^ ^® regarded as functions of x. The differential equation of the r^"^ order is found from (48) by substituting therein, after the differentia- tion, for c its value in terms of x^ y,y^, ... 2/„_i, given by (47). The result assumes the form yn = {^>y^yi'--yn-i) (49). Hence, we have |j^^in(49)=fm(48)x^in(47), . or, representing fix, y, y^...y^_^y c), by/, ^yn i^ fm = (^\^^. dy^_^ \dxdc) ' do' Hence, /^»-=i-log%- (50), ^yn-1 dx ° dc ^ provided that the first member be obtained from the dif- ferential equation, and the second member from one of its ART. 10.] SINGULAR INTEGRALS. 233 first integrals involving c as arbitrary constant. It is to be borne in mind that in effecting the differentiation with respect to X in the second member, we must regard y, y^^ ... y^_^ as functions of x. Now reasoning as in Chap. viii. since a singular solution makes 'J'~^ = 0, it makes its logarithm, and in general the differential of its logarithm, infinite. Thus we arrive at the following conclusion. A singular integral of a differential equation of the n^^ order will in general satisfy the condition ^ ' = oo , and a relation which satisfies both this condition and the differential equation will be a singular integral. Ex. 2. Applying this method to the equation, x^ y-^yx^^V',- (yi - ^yj" -y!=^^. we find, on differentiating with respect to y^ and y^ only, - {^ + 2 (2/1 - xy^] dy^ + )~--\-2x (y^ - xy^ - 1y\ dy^ = 0, whence y^ '^^2x{y^-xy^)-2y^^ Equating the denominator of this expression to 0, we find x^ + ^xy^ and substituting this value in the given differential equation, clearing of fractions, and dividing by x^ + 1, which will present itself as a common factor, IQx^y + 16y - Sx^y^ - IQxy^ - IQy^ + «* = 0, a singular integral. The equation given and the result agree with those of Ex, 1. Z.j»4« EXERCISES. EXERCISES. 1. dhi ax 2 d^y _ -a Adt ^^^' {2ax-x'f' < 8.. d'y_ 1 dx^ fj{ay) ' [CH. X. ^ 4 ^ + 1^=0. - «?'y ^y dj? X < The two following are reducible to Clairaut's form. 8 ^_ ^-f(^\ dx dx^ ^ \dafj' V<^">f'V c?i32^ dx'^ \\dxj dx^) 10. Describe the different kinds of homogeneity in differ- ential equations, and explain their connexion. The two following homogeneous equations are intended to be solved by the method developed in Art. 3. 12. .^^p-^-a?^=^o?(f]-iy\ dx dx \dxj "^ CH. X.] EXERCISES. 235 d^ii dy 13. Shew that the linear equation -7^ + ^-1^+ Qy = 0, belongs to one of the homogeneous classes, and is reducible to an equation of the first order by assuming y = e ' *. 14. Solve the linear equation -—-^ + P -^ + -^ y = 0. 15. Mainardi has remarked (Tortolini, Vol. I. p. 76), that Liouville's equation Art. 7, becomes integrable if multiplied by the factor ( -^ j . Applying this method, deduce the com- plete primitive. 16. Liouville's equation may also be solved by suppressing the second term and regarding the arbitrary constant in the first integral of the result as an unknown function of x. 17. Shew that the equation ^A+P -r + Q (-^) = is integrable in the following cases, viz. 1st, when P and Q are both functions of x, 2ndly, when they are both functions of y, 3rdly, when P is a function of x, and Q a fuuction of y. -^ >-.. fdy\^ dx d^x 18. Given U^l=a^^. 19. Given5 = V(^^ + 2/^)- (Transform to polar co-ordinates.) 20. Given s — a^. Determine the relation between y ax d'V and X, so that when x=0,-^q may have y = 0, and -r — ^- 21. Equations homogeneous with respect to x, y, and 5 can be integrated by the assumption x= e^,y = e^a. ds d^s • • • 22. Given ;;- + 3y ;t-2 = 0, required the complete primitive relation between x and y. 23. s = V(^' + 2c^). 24. s = \/(2/^ + ^^^)- 236 EXERCISES. [CH. X. 25. Examine the solution of Ex. 24, when m = 1 and when m = 0. d^x _ jjL k fdxV ' 'd?~~x'^x'\di) ' dx^ 27. Shew that x -~ is an exact differential coefficient. 28. Shew that y' + {2wy - 1) ^ + -^ ^ + x'^, = is an exact differential equation, and deduce a first integral. d II €1 ij 29. The equation -y^ + / i 2^ 2 = ^ becomes integrable by means of the factor ^x^ ~ — 2xy. (Moigno, Tom. ii. p. 672.) Deduce hence a first integral. 30. Deduce also the complete primitive. 31. Find a singular integral of the equation \dx^} X dx dx^ 82. Hence deduce a singular solution of the given differ- ential equation. 33. The complete primitive of the differential equation of the second order in Ex. 31 is required. 34. A first integral of the differential equation of the x^ second order y-xy^ + -^ y^ - {y^ - xy^'' -y' = is y + l-—a^]x^'— (1 — 2a) xy^ — a^ — y^ = 0, where y^ stands for -^ . Hence deduce the singular inteojral. Shew that it ao-rees, dx . o ' and ought to agree, with the result obtained in Art. 10. 35. Shew that the complete primitive of the above differ- ential equation is y = ^x^ + bx + 0^ + If. CH. X.] EXEECISES. 237 36. The singular integral of the differential equation of the second order, above referred to, has been found to be 16 (1 + £c') 2/ - 8ic'y, - 16x7/^ + x^- IQyl = 0. Ex. 2, Art. 10. Shew that this singular integral has for its complete primitive (16j/ + 4^' + x')i = £c (1 + x^f^ - log {(1 + x'f '-x]+7i, h being an arbitrary constant — and that this is a singular solution of the proposed differential equation of the second order. 37. The same singular integral has for its singular solution 16y + 4 The first side multiplied by dx is an exact differential and gives dy whence again integrating ?/' + aj' = 2c^ + c' (3), the equation of a circle whose centre is on the axis of x. If the direction of the radius of curvature be opposite to that of the normal, it will be necessary to change the sign of the first member of (1) . Instead of (2) we shall have ^s-(ir-i=o W' and this equation not containing x, we may depress it to the dii first order by assuming --r—V' "^^^ transformed equation is pdp dy whence ^-^^ = y ART. 4.] GEOMETRICAL APPLICATIONS. 243 Substituting for p its value -^ , we find on algebraic solu- tion cdy dx whence x= c' + clog [y + (3/^— c^y] (5). This equation, reduced to the exponential form y = i(e»+e '), (6), is seen to represent a catenary. The solution therefore indicates a circle when the directions of the radius of curvature and of the normal are the same, but a catenary when they are opposed. The latter curve has, however, many properties analogous to those of the circle. (Lacroix, Tom. 11. p. 459.) Ex. 2. To find a curve in which the area, as expressed by the formula fydx, is in a constant ratio to the correspond- ing arc. We have 2/=0(l+/)^ which, agreeing in form with the last differential equation of the preceding problem, shews that (5) represents the curve required, and connects together the properties noticed in the last two examples. Ex. 3. Required the class of curves in which the length of the normal is a given function of the distance of its foot from the origin. The differential equation is y{i+py=f{^ + yp) (1). and it belongs to the remarkable class discussed in Chap. vil. Art. 9, where the complete primitive is given, viz. f+(x-ar^{f(a)Y (2). This represents a circle whose centre is situated on the axis of X at a distance a from the origin, and whose radius is equal to 16—2 244 GEOMETRICAL APPLICATIOXS. [CH. XI. /(a). It is evident that this circle satisfies the geometrical conditions of the problem. But there is also a singular solution, found by eliminating the constant a between (2) and the equation derived from (2) by differentiation with respect to a, viz. x-a+f{a)f'{a)=0 (3). For instance, iif{ci) — n^a^ we have to eliminate a between the equations if + (x — ay = na, 2 (x — a) -\-n = 0, from which we find the equation of a parabola. While in this example the com- plete primitive represents circles only, the singular solution represents an infinite variety of distinct curves, each originat- ing in a distinct form of the function /(a). Other illustrations of this remark will be met with. The above problem was first discussed by Leibnitz, who did not, however, regard its solution as dependent upon that of a differential equation, but, establishing by independent con- siderations the equation (2), which constitutes in the above mode of treatment the complete primitive of a differential equation, arrived at a result equivalent to its singular solu- tion by that kind of reasoning which is employed in the geo- metrical theory of envelopes. Indeed it was in the discussion of this problem that the foundations of that theory were laid (Lagrange, Calcul des Fonctions, p. 268). 5. A certain historic interest belongs also to the two fol- lowing problems, famous in the earlier days of the Calculus, viz. the problem of ' Trajectories ' and the problem of ' Curves of Pursuit.' These we shall consider next. They will serve to illustrate in some degree the modes of consideration by which the differential equations of a problem are formed when a mere table of analytical expressions suffices no longer. ART. 5.] TRAJECTORIES. 24^5 Trajectories. Supposiug a system of curves to be described, the different members differing only throiigh the different values given to an arbitrary constant in their common equation — a curve which intersects them all at a constant angle is called a tra- jectory, and when the angle is right, an orthogonal trajectory. To determine the orthogonal trajectory of a system of curves represented by the equation ^{x, y, c)=0 (1). Representing for brevity ^ {x, y, 6) by ^, we have on dif- ferentiating ~ dx+ -r dy= 0. ax ay Hence, for the intersected curves, dy _ d(f> dcf) dx dx ' dy ' IN'ow representing this value by m, and the corresponding value of ~^ for the trajectory by rn, we have, by the condition of perpendicularity, m = . Hence for the trajectory dy d(^ d(j) dx dy ' dx' 'p^-fjy=' ^^)v which must be true for all values of c. Hence the differential equation of the orthogonal trajectory will he found by elimi- nating c between (1) and (2). Were the equation of the system of intersected curves pre- sented in the form (j> (x, y, a, &) = 0, a and h being connected by a condition ^(a, &) = 0, 246 TRAJECTOEIES. [CH. XI. we should have to eliminate a and h between the above two equations, and the equation d^ {x, y, a, h) ^^ dcj) (x, y, a, h) . ^ ^^ dy dx We shall exemplify both forms of the problem. Ex. 1. Required the orthogonal trajectory of the system of curves represented by the equation y = ex". Here ^ = ?/ — cx"^, whence by (2) dx + ncx''~^ dy = 0. Eliminating c, xdx + nydy = ; therefore x^ + ny"^ =c , the equation required. We see that the trajectory will be an ellipse for all positive values of n except n = 1, — an ellipse, therefore, when the intersected curves are a system of common parabolas. The trajectory is a circle if n = \, the intersected system then being one of straight lines passing through the origin. The trajectory is an hyperbola if n is negative. Ex. 2. Required the orthogonal trajectory of a system of confocal ellipses. The general equation of such a system is 2 2 2 ~ 72 •*■) a- W a and h being connected by the condition where li is the semi-distance of the foci, and does not vary from curve to curve. Hence we have to eliminate a and h from the above equations, and the equation j-^Ax 2 (^y = : ART. 6.] TRAJECTORIES. 247 the result is the solution of which may be deduced from that of Ex. 3, Chap. VII. Art. 10, by assuming therein A = lj B = h^. We find X2 2 9 9 9 / 1 G ^ c' + 1 ' and this may be reduced to the form 2 2 a^ and \ being connected by the condition Thus the trajectory is an hyperbola confocal with the given system of ellipses. 6. When the trajectory is oblique, then 6 being the angle which it makes with each curve of the system, and m and m' having the same significations as before, , m + tan 6 1 — m tan or, substituting for m its former value — ^ ^ -^ , and for 7n ax ay its value -~- as referred to the trajector}^, we have on reduc- tion ay _ ay ax dx d(b d(b , ^ ^ ^^ -^ + -^ tan ^ ay ax an equation from which it only remains to eliminate c by means of the given equation in order to obtain the differential equation of the trajectory. Ex. Required the general equation of the trajectories of the system of straight lines y = ax. 248 TEAJECTOEIES. [CH. XI. Here (l> = y — ax, whence by (3) di/ tan 4- a dx \ — a tan d X tan 6 + y x — y tan ' or (3/ + ^ tan 6) dx + (3/ tan 6 — x) dy = 0, a homogeneous equation, an integrating factor of which being —r, r> , we have ydx — xdy , ^ xdx + vdti ^—^ ^ + tan 6 — ^ — ^ = 0, x^ + y' x' + y" ' whence integrating X tan ^ - + tan Q log {x^ + 3/^)^ = c. If we change the co-ordinates by assuming x = r cos (^, y = r sin , we get the equation of a logarithmic spiral. The following example, which is taken from a Memoir by Mainardi (Tortolini's Annali di Scienze Matematiche e Fisiche, Tom. I. 251), is chiefly interesting from the mode in w^hich the integration is effected. Eequired the oblique trajectory of a system of confocal ellipses. Representing the tangent of the angle of intersection by n, we have to eliminate a and b between the equations i + ^ = l, a'-V^V, a y ^ i^ = - y X ART. 6.] TRAJECTORIES. 249 The result may be expressed in the form [nx + y-{-(ny — x) p] [x — ny + {nx ■\-y)p] = h^ (n -p){l + np). To integrate this equation let us assume a) — ny-{- (nx + y)p = M(l + np), M {nx + y+ (ny — x)p} = h^ (n—p). As these on multiplication reproduce the given equation the assumption is legitimate. Eliminating p from the last two equations, and dividing by 1 + n^, we have {x'+y'-\-h')M=x(M' + h') (a). Differentiating this equation and eliminating y and p from the result by the aid of any two of the last three equations (it is evident that two only are independent), we obtain a differential equation between M and x, which is capable of expression in the form nd jxM) X ^ ^ {h:'{xM)-{xMy]^ M/ _M\i "^ '' x \ xJ [Far {a) may be written thus : Mf=^{x-M){W-xM) (c); differentiating we have '^ dx dx dM\„, ,^, , .rJ.^. dM\ therefore ^/J^ ^^i- - M){h^ -.M)dM ^ dx M dx = 1- 250 therefore TEAJECTORIES. [CH. XI. ^ dy ^r ^ (h - xM) dM ^ doo M dx therefore But therefore dy M -\-ny — x dx n[x — M) + y ' 2My{M-x)-\-2nMy'' therefore = n{x-M) k' -2xM -\- M' -- ^{h' - M')'^\ ; therefore {M' - h') (y - ^^) -n{x-M) {M' - h') ^n{x-M)^JM^-¥)'^^ M' dx ' therefore yxdM , ,,. , ,--, X dM M dx dM therefore n{x- M) (M + x--j-\-\'y(x^ - ilf ) = therefore n[x— M) — ^ — - + yx dx X ART. 7.] TRAJECTORIES. 251 Hence by the aid of (c) we obtain (J).] Hence, by integration in which it is only necessary to substitute for itf its value in terms of ^ and y deduced from [a). Curves of Pursuit, 7. The term curve of pursuit is given to the path which a point describes when moving with uniform velocity towards another point which moves with uniform velocity in a given curve. Let X, y be the co-ordinates of the pursuing point, x\ y the simultaneous co-ordinates of the point pursued. Also let the equation of the given path of the latter be /(^',2/') = (4). Now the point pursued being always in the tangent to the path of the point which pursues, its co-ordinates must satisfy the equation of that tangent. Hence, 2/'-2/ = x!(^'-^) • ^^)- dx Lastly, the velocities of the two points being uniform, the corresponding elementary arcs will be in the constant ratio of the velocities with which they are described. Hence, if the velocity of the pursuing point be to that of the point pursued as ?^ : 1, we have n sj[dx^ -f dy'^) = ^J^dx' + dy'), or, taking x as independent variable, 252 CURVES OF PUKSUIT. [CH. XI. the sign to be given to each' radical being positive or negative, according as the motion tends to increase or to diminish the corresponding arc. From (4) and (5), when the form of the function / (a?', y) is determined, x and y' may be found in terms of x, y, and -j- , and these values enable us to reduce (6) to an equation be- (a/IJ du'u tween x, y, -~ , -r^ . It only remains to solve this differ- ential equation of the second order. If the signs of the radicals are both changed, the motion in each curve is simply reversed, and the curve of pursuit becomes a curve of flight. Bat the differential equation remaining unchanged, the forms of the curves are unchanged, and only their relation inverted. Ex. A particle which sets off from a point in the axis of a), situated at a distance a from the origin, and moves uniformly in a vertical direction parallel to the axis of y, is pursued by a particle which sets off at the same moment from the origin and travels with a velocity which is to that of the former as ?i :1. Required the path of the latter. The equation of the path of the first particle being x =a, (5) becomes whence Thus we have dy , . y =y^{a-x)j-^. dx _ ^ dy _ ( \ ^y dx ' dx dx^ * and the differential equation, both radicals being positive, is dx'^ y T ^\dx) «(-«)S=./ii + flT[ («)■ ART. 8.J CURVES OF PURSUIT. 253 Hence dx' n , MyYI n{a-x) Multiplying by dx and integrating ..v^-e)}=^^"-^^ (J). therefore -~ = -^{c{a — x) *" (a — ^)"}. Hence, if n be not equal to 1, _i 1 1 (c(a- xf " 1 (a - ^) "] n n But if n be equal to 1, we liave dy _\ ix— a c \ dx 2 1 c x — ay whence y = ^-^ --log(x-a)+c. 8. The class of problems which we shall next consider is introduced chiefly on account of the instructive light which it throws upon the singular solutions of differential equations of the second order. Inverse Problems in Geometry and Optics. The problems we are about to discuss are the following : 1st 5 To determine the involute of a plane curve. 2ndly, To determine the form of the reflecting curve which will produce a given caustic, the incident rays being supposed parallel. 254 PROBLEMS [CH. XI. In both these problems we shall have occasion in a parti- cular part of the process to solve a differential equation of the first order of the form y-x4. ip) =//- ./. (i>) - ^ ip) /'-' (p) (7), in which the independent variable. Let us then assume (j) (p) = V, and transform the differential equation so as to make x and v the new variables. Substituting v for {p) in (7), we have y-xv=ff'-\v)-vf'-\v) (8). Differentiating, and regarding v as independent variable, dv dv dv^ ^ ^ -^ ^ ^ dv ^ ^ But di/ dx ,.if-.dx AET. 8.] IN" GEOMETRY AND OPTICS. 255 Hence, {-\v)-v]f^-x^-r'(v}, or, dx X _ f'~^ (v ) dv V — (jT^ {v) V — (pr^ (y) ' Hence, if for brevity we write /.-rf^=t(^) • <^>' we have X = e-'^(^) [C + e'^(^) fXv)f-' {v) dv] = e-^(^y [G + 6'^(«)/-^ (v) - fe^^^) df-' (v)}, whence x-f-'(v) = e-^<^^^G-j€^'^^)df-'(v)] (10), between which and (8), v must be eliminated. If in those equations we make f'~^ (v) = tj they assume the somewhat more convenient form, y-xf'{t)=f{t)-tf'{t), aud these may yet further be reduced to the form fit) ^ -* = -^9#=^^n^ (11). From these equations it only remains to eliminate t, the forms of / and cf) being specified, and that of yjr given by (9) ; and this is apparently the simplest form of the solution. 256 PROBLEMS [CH. XI. 9. We shall now proceed to the special problems under consideration. To determine the involute of a plane curve. It is evident from the equations which present themselves in the investigation of the radius of curvature, that if a?, y be the co-ordinates of any point in a plane curve, and x ^ y those of the corresponding point in the evolute, then X =x — '■-^ ^-^ , V = y -\- — — ^— , q ' -^ / ^ ' where Jp = -r- , q = -A (Todhunter's Differential Calculus, Art. 320). Hence, if the equation of the evolute be 2/' =/(«'') (12), we shall have on substituting therein for y and x the values above given, ,^l±P!./j._Wl±i5} ..(13), a differential equation of the second order connecting x and y, and therefore true for each point of the curve whose evolute is given. Of that evolute the curve in question is an involute. Hence, if y z=f{x) be the equation of a given curve, the equation of its involute will satisfy the differential equa- tion (13). Now suppose that nothing was known of the genesis of the above equation, and that it was required to deduce its complete primitive, and its singular solution, should such exist. Upon examination the equation (18) will prove to be of a kind analogous to that of Chap. VII. Art. 9. If we assume .-^M^=„ q ART. 9.] IN GEOMETRY AND OPTICS. 257 a and h being arbitrary constants, we shall find that each of these leads by differentiation to the same differential equation of the third order, viz, Zp^^^{l+f)r=0 (16), where r stands for ~. . It follows hence, that a first integ^ral of (13) will be found by eliminating q between (14) and (15), and connecting the arbitrary constants h and a by the relation h =f{cb)» Eliminating q, we find x-a + {y-l)p^O (17), wherein making h =f{a), we have ^-<^ + {y-f{o)]p = o (18), for the first integral in question. Again, integrating, we have ix-af+{y-f{a)f^r' (19), in which-^ and r are arbitrary constants. This is the complete primitive of (13). It is manifest from its form that it repre- sents, not the involute of the given curve, but the circles of curvature of that involute. Indeed, that the complete primi- tive cannot represent the involute might have been affirmed a priori. The equation of the involute of a given curve cannot involve in its expression more than one arbitrary constant ; for the only element left arbitrary in the mechanical genesis of the involute is the length of a string. It remains to examine the singular solution of (13). This is most easily deduced by eliminating a between the first integral (18) and its derived equation with respect to a, viz. between the equations x-a + {y-f{a)]p=^0 (20), -l-f{a)p=0 (21). From the second of these we have "/^■(t)- P B. D. E. 17 258 PROBLEMS [CH. XI. Hence eliminating a from (20) x+i^P =/'- (:^) +^//'-' (zl) (22), wliicli is the singular solution of (13), and the differential equation of the fii'st order of the involute sought. This equation is a particular case of (7). If we express it in the form »-(T)'=-«'-(T)-(T)^-(y)- we see that it is what (7) would become on making Hence comparing with the general solution (11) we have tW=/-^ = log(t^^ + l)^. V Thus the system (11) becomes .-..^±h£^ w. The final solution is therefore expressed in the following theorem. Given the equation of a curve in the form y ^f{x'), that of its involute is found by eliminating t from the system (23). 10. Parallel rays incident, in a given direction, on a reflect- ing plane curve produce after reflection a caustic whose equa- tion is given. The equation of the reflecting curve is required. ,2> ART. 10.] IN GEOMETRY AND OPTICS. 259 Let IP be a ray incident parallel to the axis of a? on a point P in the reflecting curve 8PM, Fig. 1, PP'Q the reflected ray cutting the axis of ic in Q and touching the caustic S'P'M' in P'. Let X, y be the co-ordinates of P, x\ y those of P'. Let the equation of the caustic be y =f{x). It is an easy consequence of the law of reflection that the angle PQX which the reflected ray makes with the axis of x is double of the angle PTX made by the tangent at P with ^ the axis of x. This at once gives us the equation y-y _ 2p X^ X 1—p where V — ^^- Hence y-y'-^^{^-^)=^ (24). As, however, {x ^ y') is a point at which consecutive re- flected rays intersect, we are permitted to difierentiate the above equation regarding x and y' as constant while x and y vary. We thus obtain, representing -7^ by g, or whence x — x' = — ^ , zq and x' = x + ^^^^^'^ (25). Substituting this value in (24), we have whence y—yj\.£~ (26). 17—2 260 PROBLEMS [CH. XI. Were the equation of the reflecting curve given and that of the caustic required, it would only be necessary to substitute in (25) and (26) the values of^and q^ in terms of a?and ^derived from the former, and then by eliminating x and y from the three, to deduce the relation between x' and y . Conversely, to determine the reflecting curve we must elimi- nate X and y from (25), (26) and the equation of the caustic,' viz. y' =f{x). The result which is obtained by mere substi- tution is ^-f/H"-V^} (27), a differential equation of the second order, the solution of which will determine in the fullest manner the possible rela- tions between x and y which are consistent with the conditions of the problem. Were this equation given and nothing known respecting its origin, we might at once infer that it is of a class analogous to those of Chap. vii. Art. 9. For writing y^i=^' -+^-V^=<^ (28), we find that each of these leads by differentiation to the same differential equation of the third order. For the first gives while the second gives ,2 2^ , 'Zq^ and these lead to the same value of the differential coefficient of the third order, r, viz. this constituting the essential criterion of agreement between differential equations of the third order. ART. iO.] IN GEOMETRY AND OPTICS. 261 Accordingly, eliminating 5- from (28) and afterwards making h —fia) by virtue of (27), we find p _ 1 -p ^ f{a)-y~f{ar-^)' ± jj which is a complete first integral of (27). We see that it agrees, and necessarily so, with (24), a only taking the place of x and f{a) that oiy. The complete integral of (29) will be found to be {2/-/(a)f =4m(aj-a) + 4m' (30), m being an arbitrary constant. And this is the complete primitive of (27). If we substitute x for a, which we may without loss of generality do, then f{a) =f{x) =y', so that the above equation gives {y —yy = 4 = s and ATS-=(j). From this equation the ordinary equation in rectangular co- ordinates may be found in the following manner. Still taking the beginning of the arc as origin, let the tangent at that point be taken as the axis of x, then will the element of the curve ds be inclined at an angle fX) ddii^\ ' /{i.g}]^.. =/(,)., ART. 11.] INTRINSIC EQUATION OF A CURVE. 265 whence il + (^V dx=f(cj,) d^ ...(38). di/ But, since -^ = tan (p, the above becomes sec (j>dx —f {(f)) d(j), dx = cos (pf (cj)) d(f), X = j cos (f)f' ((j)) dcf), and in like naanner employing for 5 the equivalent formula y = jsmcj>f{(j))d(l), which agree with the previous expressions. Another consequence should also be noted. From (38) we 'dy\\^ r, / , X d^ '"•{'^e)}"-^*^^- A But -^ = -j- tan~^ 1^1= j — 2 ; whence ax ax \dxj - (dyv ' d^j dxj j J ^T^{ ^dyV ' \dx) Therefore S—^L^=f'{j>). dx^ Now the first member being the expression for the radius of curvature p of the given curve, we have ?=/'(= 6j and the intrinsic equation becomes s = G- — ^^— ^ {€"**- 1). m From this it appears that any intrinsic equation of the form s = a{e'^^-l) (40) will represent an equiangular spiral. Given the intrinsic equation of a curve, to deduce that of itsi'^^ evolute. Considering the given curve as formed by the unwinding of a string from its evolute, any arc of the former may be said to correspond to that arc of the latter by the unwinding of the string from which it is formed. Thus if s, ' represent elements of the evolute corresponding to s, in the given curve, then the origin of s is that point of the evolute whose tangent forms the radius of curvature at the origin of 5. This premised, it is evident that we shall have ART. 12.] INTRINSIC EQUATION OF A CURVE. 267 For the extreme differential elements of the arc of the evolute are respectively perpendicular to the corresponding extreme differential elements of an arc of the given curve. Hence the inclination of the former being equal to that of the latter, the value xrf-^ 4s4Jib same for both. Secondly, any arc of the evolute is by a known property equal to the difference of the radii of curvature of the ex- tremities of the corresponding arc of the given curve. Hence if pQ represent the radius of curvature at the origin of the given curve, we shall have s' = p-Po=/'W-/'(0). by (39), and, substituting cj)' for (j), s'=/(<^')-/(0). Dropping the accents, we may therefore affirm that if the intrinsic equation of a curve is s =/{) t that of its evolute willbe5=/((/>)-/X0). Ex. The intrinsic equation of the logarithmic spiral is s==a {e^'^ — 1). Hence that of its evolute is s = mae'^^ — ma • =- ma (e"**^ - 1), which also denotes a logarithmic spiral. Given the intrinsic equation of a curve in the form s =/() wherein /(^) vanishing with ^ is supposed capable of expan- sion in the form f{4>)^A,4.+A,^' + A,' + &c (41), required the general intrinsic equation of the involute. As to any curve there belong an infinite number of invo- lutes depending on the different values given to that initial tangent to the curve which forms the initial radius of curva- ture of the involute, we shall represent the arbitrary value of that initial tangent by G, Now if 5 = i^ ((/>) be the intrinsic equation of the involute, we have by the last proposition F'{). 268 INTRINSIC EQUATION OF A CURVE. [CH. XI. But i^'(O), being the initial radius of the curvature of the in- volute, is equal to C. Hence the above equation may be expressed in the form when ce F ((p) = [ f (cf,) d<}> + G

)d(f> + a(P (42). If, for distinction's sake, we represent the arc of the invo- lute by Sj the equation may be expressed in the form s'=j(a + s)d(l> (43). It is to be remembered that the lower limit of the integral isO. The following proposition from the memoir of Dr Whewell referred to, will illustrate the application of the above theo- rems. Let any curve be evolved, and the involute evolved, and the involute of that evolved, beginning each evolution from the commencement of the curve last formed, and with a "rec- tilineal tail" which is of constant length for all. The curves tend continually to the form of the equiangular spiral. Let s, s', s"y &c. be the successive curves, <^ the angle which is the same for all, and let the tails represented in fig. 3, by AA', A'A'\ A" A'", &c. be each equal to a. ART. 12.] EXERCISES. 269 Then representing the equation of the given curve by s =:f (^), we have for the first involute the equation / =l(a + s) d(j> = acj)-{- jf(cl))d^, and in general Now giving to/((^) the form (41), we have /Vw# 1.2.. .(7^ + 1)^1. 2.. .(71+2)^ We see then that the first n terms of the expression for 5^*"' in terms of (j> are unaffected by the form of the function f {(f>)y while those which remain are affected with coefiicients which tend to 0. Thus the limiting form of (44) becomes -«^+12^1.2.3"*^ • = a(6^-l) , (45). Now this is the equation of an equiangular spiral. EXERCISES. 1. Determine the curve whose sub tangent varies as the abscissa. 2. Determine the curve whose normal varies as the square of the ordinate. 3. Shew that the curve in which the radius of curvature varies as the cube of the normal is a conic section. 270 EXERCISES. [CH. XI. 4. Find a curve in which the length of the arc is in a constant ratio to the intercept cut off by the tangent from the axis of cc. 5. Shew that the above is a particular case of curves of pursuit. 6. Find the orthogonal trajectory of a system of circles touching a given straight line in a given point. 7. Find the orthogonal trajectory of the system of ellipses 2 2 defined by the equation -a + t^ = 1, h being the variable parameter. 8. Find the equation referred to polar co-ordinates of the curve in which the radius vector is equal to n times the length of the portion of the tangent intercepted between the point of contact and a straight line drawn from the pole to meet the tangent at a given angle. 9. E-equired the form of a pendant in Gothic architecture supposed to be a solid of revolution, such that the weight to be supported by each horizontal section shall be proportional to the area of that section. 10. Required the curve in which s — ax^. 11. A curve is defined by this property; viz. that the radius of curvature at any point is a given multiple {n) of the portion of the normal intercepted between the point and the axis of absciss86; prove that the length of any portion of the curve may be finitely expressed in terms of the ordinates of its extremities. (Cambridge Problems, 1849.) 12. Find a differential equation of the first order of the curve whose radius of curvature is equal to n times the nor- mal, and shew that this is always integrable in finite terms if n be an integer. 13. Shew that if w = 2 the curve is a cycloid, if w = 1 a circle, if n = — 1 a catenary. 14. The curve whose polar equation is r*" cos mO = a"* rolls on a fixed straight line. Assuming that straight line as the CH. XI.] EXERCISES. 271 axis of X, shew that the locus of the carve described by the pole of the rolling curve will have for its equation 2m (Frenet, Eecueil d'Exercices sur le Calcul Infinitesimal.) Note. To solve problems like the above, we observe that if RTS, Fig. 4, represent the given curve rolling on the given line OX, and APC the curve described by the pole P, then taking OX for the axis of x, and putting OM—x, MP=y, the straight line PT joining that pole with the point of contact will be a radius vector of the given curve, but a normal of the described curve. Hence r = vi^ - m\ '^'- Again, PM is the perpendicular let fall from the pole upon the tangent of the given curve, but the ordinate y of the required curve. Hence yJ{dr^ + rHd'^) y ^'' By means of [a], (b), and the equation of the given curve, eliminating r and d, we obtain the differential equation of the curve sought. 15. In the particular case of m = J the rolling curve will be a parabola, the pole its focus, and the described curve a catenary. 16. If m = 2, the rolling curve is an equilateral hyperbola, the pole its centre, and the described curve an elastica. ( 272 ) CHAPTER XII. -cA- p^ ORDINARY DIFFERENTIAL EQUATIONS WITH MORE THAN TWO VARIABLES. 1. The class of equations whicli we shall first consider in this Chapter, is represented by the typical form, Pdx+ Qdy+Rdz=-0 ...(1), P, Q and R being functions of the variables x, y, z ; and it is usually termed a total differential equation of the first order with three variables. Possibly the first observation suggested by the examination of this form will be, that it does not answer to the definition of a differential equation, as the expression of a relation in- volving differential coefficients, Chap. i. And certainly it does not exhibit their notation. If, however, we attempt to attach a meaning to the general form (1), we shall perceive that the idea of a limit is involved essentially. And if we study its origin, we shall see that this idea may be expressed, here as elsewhere, in the language of differential coefficients. For (1) is not understood as implying simply that the expression, P^x+ QAy + BAz (2), approaches to the value when the increments Ax, Ay, Az approach that value, true though it be that the vanishing of the increments causes that expression to vanish with them. But what (1) is always understood to express is, that in the approach to the limiting state, (2) tends to vanish in conse- quence of the^m^i05 which the increments Ax, Ay, Az tend to assume; it is, that if we represent (2) in any of the equivalent forms PAx + QAy + RAz PAx + QAy + RAz ^ . Ax "^^^ Ay ^y^ ^''' the limit of the ratio expressed by the first factor of each is 0. And the problem of the integration of (l);is that of the discovery ART. 1.] EQUATIONS WITH MOEE THAN TWO VARIABLES. 273 of the possible relation or relations among the primitive vari- ables which will secure this result, supposing Ax, Ay, Az to be so restricted as to preserve such relations unviolated. Now whether the primitive variables are connected by one equation or by two simultaneous equations (we cannot sup- pose them connected by three equations without making them cease to be variable), the relation (1) is fully expressible in the language of differential coefficients. If there exist one primi- tive relation which, as we shall hereafter see, can only happen under particular circumstances, then , dz ^ dz 1 az — -J- dx-\--^r- dy, dx dy *^ while (1) is presentable in the form dz — — pj Jic — -^ dif. Hence, since dx and dy are independent, we have dz _ P dz _ Q , Tx~~R' dy'^B ^^^' a system which in the supposed case is equivalent to (1). On the other hand if, as will usually happen, two simultaneous equations connect the primitive variables, e.g. 4>(x,y,z) = 0, ylr(^x,y,z) = (4), then, since we have ^^-dx + ^-^du-\--i^dz-0 dx'^'^^dy'^^^ dz"^'-^^' ~- dx -f -J- dy + -J- dz = 0, the elimination of dx, dy, dz between these and the original equation gives p /'d(j> dyfr d dyjA ^ fd^ d-^jr d(j) d^^r^. \dy dz dz dy) \dz dx dx dz J + E(p^-f^)=0 (5), \dx dy dy dx) ^ ^ B. D. E. 18 274 ORDINARY DIFFERENTIAL EQUATIONS [CH. XII. a result which is equivalent to (1), but is expressed in the language of partial differential coefficients. As it cf>nstitutes but a single relation between two unknown functions ^ and -v/r, one of the two may be considered arbitrary, and a particular form being given to it, we should have a partial differential equation for determining the other. We propose indeed to discuss the equation (1) under its actual form, but it is not unimportant to shew that it con- stitutes no real exception to the definition of a differential equation. Treated by the methods proper to partial differ- ential equations, the forms (3) and (5) lead to the same solutions as those investigated in this Chapter. 2. The foregoing remarks admit of geometrical illustrations. If X, y^ z and x + Ao?, y + A^/, z -\- lS.z are the co-ordinates of two points, the value of the expression PAic + Q Ay + R/^z, where P, Q, R are given functions of x, y, z, will depend solely upon the positions of the points. If we suppose the second point to approach the first along any path, the value of the above expression will approach to 0, in consequence of the quantities Aic, Ay, A^ approaching to 0, and independently of the ratios which they assume in vanish- ing. But this is not in accordance with the understood meaning of the equation (1). The increments therefore not being independent, either they are connected by one relation, in which case one point being given the other must lie on the surface which that relation determines, and its approach to the first must be made along that surface, but is in no other way restricted ; or the incre- ments are connected by two relations, and then, the first point being given, the second must be on the line determined by those relations, and its approach to the first must be made along that line, and therefore in a definite path. 8. These considerations suggest to us the following ques- tions for analysis, viz. : 1st. Under what circumstances is the solution of the equa- tion Pdx -{■ Qdy + Rdz = expressed by a single relation be- tween the primitive variables — a. relation which with the AKT. 3.] WITH MORE THAN TWO VARIABLES. 275 arbitrary constant of integration will represent a family of surfaces ; — and how is such a relation to be determined ? 2ndly. How is the solution to be obtained when the above condition is not satisfied ? These questions we shall next consider. The equation Pdx + Qdy + Rdz = 0, derivable from a single 2?rimitive. From the given equation, we have dz = — -^dx — -^ dy (6). But the existence of a single primitive involves the sup- position that 2; is a function of x and y, and therefore that we have dz _ P dz _ Q dx~~M* dy^^R ^"^• P Q Hence, if -^ and -^ do not contain z, we have, by the property of differential coefficients, S^-^*-*- ^'^^^ » y ' d^F^d^Q ^^ ^ ^ dyR dxR' P Should however -^ and ~ both or either of them contain z, then, because we can still regard them as ultimately functions of X and y, for z is such by hypothesis, we must change the above into d^P dld^P_d^Q dz^d_Q dyR dydzR dxR dxdzR' uz cf z Lastly, substituting here for -7- and ^ their values given in (7), effecting the differentiations, and reducing, we have p(dQ^dR\ ^d^_d_P^ (djP_dQ\__ ""Kdz dyl^'^Ux dzl^^Kdy -dx)-^ ^^^> an equation of condition which, when identically satisfied, 18—2 276 ORDINARY DIFFERENTIAL EQUATIONS [CH. XII. indicates that the proposed equation admits of a single pri- mitive. 4. To deduce the complete primitive of the differential equation Pdx + Qdy + Rdz = when the equation of condition (8) is satisfied. The supposed primitive involving all the variables x, y, z, it is evident that if we differentiate it on the hypothesis that z is constant, we shall arrive at a result equivalent to Pdx + Qdy = 0. It is also evident that if the primitive con- tained a function of z for one of its terms, that term, whatever the form of the function might be, would disappear in the dif- ferentiation. Conversely then if we integrate the equation Pdx+Qdy=0 (9), regarding z as constant, and adding in the place of an arbitrary constant an arbitrary function of z, we shall arrive at a result which will necessarily iyiclude the complete primitive, and in v/hich it will only remain necessary to determine what form must be given to the arbitrary function of z. Thus, if the integrating factor of (9) be //<, and if, assuming z constant, we write dV dV fi {Pdx+ Qdy) =-^ ^^^~dy ^y> then will the complete primitive be of the form V={z) (10), in which it only remains to determine <^ {z). And this will be done by differentiating with respect to all the variables and comparing with the given equation. Differentiating (10) then with respect to x, y, z, and trans- posing, we have whence ART. 4.] WITH MORE THAN TWO VARIABLES. 277 Now by the given equation, Pdoc + Qdy = — Rdz. Substi- tuting, and rejecting the common factor dz, we have dz dz whence ^=i^-^ -ai)' the second member of which must, on the hypothesis that a single primitive exists, be reducible to a function of z by means of (10). The solution of the equation thus reduced will determine ^ [z), the value of which substituted in (10) will give the complete primitive. Although we are fully entitled to affirm that the equation determining [z) must, whenever a single primitive exists, be reducible to a form not involving x and y ; it may be proper to verify this conclusion a posteriori. Let us then inquire under what condition the function dV -^ yu-i^ can be freed from both x and y by means of the GiZ equation V= {z) being an arbitrary function of z introduced in the place of an arbitrary constant. Now, differentiating with respect to all the variables, we have dx-\--, — ; — ^dy — \ — ; h ^} ' \dz = Q, {y + ay ^ \y+a dz ] or {y + ay dx + zdy - \y + a -{- {y -\- ay } \ dz = 0, which agrees with the equation given, if we have dz 280 ORDINARY DIFFERENTIAL EQUATIONS [CH. XII. Here then <^{z) —c and the complete primitive is z X — - = c (a). If we commence by regarding y as constant we obtain by a first integration z-{y-\-a)x=^ (y) _ z ay y + a This equation involves both x and y^ but it is reducible by the previous one to the form dy y + a' d^ (y) _ dy or ^(j/) y + a' of which the integral may be expressed in the form h being an arbitrary constant. Hence, finally, z = {y + a) X + h {y •{■ a) = (y + a){x + h), and this is equivalent to the former result (a). Ex. 2. Given zdz + {x — a)dx = [h^ — z^-{x — d)^\^ dy. Integrating as if y were constant we have z^ ■\- (x-ay = {,) To apply this to the problem of drawing lines satisfying the conditions of the problem on the ellipsoid 2 2 2 a^'^t''^?^ ^^^' it is necessary from the above three equations to eliminate a; and t/. From the second and third which here suffice, we have whence (f) (z) = — z'^ -{- G. Therefore a)'-hy' + z''=C (/). The particular solution sought is therefore expressed by the equations (e) and (/), which are together equivalent to the previous solution expressed by (b) and (d). Total differential equations containing more than three variables. 9. It will suffice to make a few observations on the equa- tion with four variables Fdx+ Qdy + Bd^ + Tdt = ..(24), and to direct attention to the general analogy. ART. 9.] MORE THAN THREE VARIABLES. Writing the above equation in the form R 287 (Xt — —~ fj-j CfjSG T dz (25), it is evident that, in order that it should be derivable from a single primitive, we must have d\Q_(d\P^ fd_\B_fd\Q f^\P_fd\E dx)T~\dyjT' \dy)T~\dz)T' \dz) T~ \dscj T ' where [ -^ ] refers to x not only as appearing independently, but also as implicitly involved in t ; and so on for the rest. dt dt dt Effecting the differentiations, and substituting for -^ , -j- ■, -J- their values implied in (25), we have \dx dy) \dy dt j \dt dx) „fdR dQ\ „ fdT dR\ „ fdQ dT\ „ \d2 dxj \dx dt -^(f-S)=« which are the equations of condition of existence of a single complete primitive. It is evident from the symmetry of the problem that the equation must also hold here. But this is not a new condition. It may be deduced from (26), by multiplying the respective equations of that system by i^, P, and Q, and adding the results. 288 EQUATIONS WITH MOEE THAN THREE VARIABLES. [CH. XII. It is obvious that when there exist n variables, the num- (n — l)(n— 2) ber of independent equations of condition is — - — ^ , being the number of ways of equating two partial differential coefficients in a system in which 71 — 1 are contained. The solution of any such equation may be effected by an extension of the method adopted for equations with three variables. We must integrate as if all but two of the varia- bles were constant, adding, in the place of an arbitrary con- stant, an arbitrary function of the variables which remain. This function we must determine by differentiating with re- spect to all the variables, and comparing with the equation given. If a single primitive exist, such determination will be possible. If a single primitive do not exist, we must, follow- ing the analogy of the corresponding case of three variables, endeavour to express the solution by a system of simultaneous equations. And such is indeed its general form. Pfaff, in a memoir published by the Berlin Academy 1814 — 15, has shewn that, according as the number of variables is 2n or 2n -1- 1, the number of integral equations is noi n + 1 at most. His method, which is remarkable, consists of alternate inte- grations and transformations. For important commentaries and additions see Jacobi (Werke, Tom. I. p. 140), and Raabe (Crelley Tom. xiv. p. 123). Ex. Given (2x+y'^ + ^xy^—y^ dx+ 2xydy — xdy^ -I- x^dy^ = 0. If we suppose the variables y^,y^ constant, we have to in- tegrate {2x + ?/^ + 2xij^ — y^ dx -\- 2xydy = 0, which, on substituting an arbitrary function of y^y y,^ repre- sented by ^, for an arbitrary constant, gives x''-\-xy^-\-x'y^-xy^ = . Differentiating with respect to all the variables, we have (2x + 3/^ + 2xy^ — 2/ J dx + 2xydy — xdy^ -f- x^dy^ d(b , dS , AET. 10.] EQUATIONS WITH MOEE THAN THREE VARIABLES. 289 Comparing this with the given equation, we have whence ^ = c and the solution is 0^ + xy^ -{- c^y^ — xy^ = c (a). Had we begun by making x and y constant, we should have had as the result of the first integration, ^^^2-^2/1 = ^ i^)y ^ denoting a function of x and y. Differentiating with respect to all the variables and comparing with the given equation, we should find dj> = — {2x ■\-y'^dx— 2xy dy^ whence (^ = — x^-~ xy^ + c, the substitution of which in (5) reproduces the former solu- tion (a). Equations of an order higher than the first. 10. When an equation of the form Adx^-\-Bdy''+Cdz' + 2Ddydz-\-2Edxdz + 2Fdxdy^0...{2S) is resolvable into two equations each of the form Pdx + Qdy + Rdz = 0, the solution of either of these obtained by previous methods, will be a particular solution of (28), and the two solutions taken disjunctively will constitute the complete solution, which is therefore expressed by the product of the equations of these solutions, each reduced to the form F= 0. The condition under which (28) is resolvable as above, is expressed by the equation ABC+2DEF-AD'-BE'-CF' = (29). B. D. E. 19 290 EQUATIONS OF A HIGHER ORDER. [CH. XII. This is shewn by solving (28) with respect to dx, and assuming the quantity under the radical to be a complete square. Thus, the equation afdx^ + y^di/^ — z^dz^ + ^xy dx dy = 0, w^hich will be found to satisfy the above condition, is resolv- able into the two equations scdx + ydy + zdz = 0, xdx + ydy — zdz = 0, whence x^ -\-y'^ + z^ = c ..,(a)y x^ -\- y'^ — z'^ = c (Q. Geometrically the solution is expressed by lines drawn in any manner on the surface, either of the sphere (a), or of the hyperboloid (6). When the condition (29) is not satisfied, the proposed equation does not admit of a single primitive, or of any dis- junctive system of primitives. But it does in general admit of a solution expressed by a system of simultaneous equ.ations. Thus, if we integrate • the equation dz^ = m^ {dx^ + dy^) , sup- posing X constant, we find z = my + (7, or, replacing by a function of x, Zi= my + {x) the system (c) {d) will therefore constitute a solution of the equation given. We enter not into the question whether it is the most general solution or not, proposing merely to exem- plify the kind of solution of which the equation admits. To this we may add that all equations which do not satisfy the conditions of integrability, though they may present themselves in the form of ordinary, have a far more intimate connexion with partial differential equations ; and that this connexion affords the best clue to the solution of their theo- retical difficulties. CH. XII.] EXERCISES. 291 EXEECISES. X — a y — h z — o 2. {x-'^y-z)dx\ {2y-'^x)dy+{z-x)dz = (). S. {y ^z)dx+{z-\- x) dyA-{x-\r y) dz = 0. 4. yz dx + zxdy + xydz = 0. 5. (y -{• z) dx + di/ -\- dz = 0, 6. ay^z'^dx + hz'^x^dy + cx^y'^dz = 0, 7. (^^2/ ~ 2/^ "" 2/^^) <^^ + {^y^ ~ ^^^ ~ ^^) ^y + {^y'^ + ^^y) ^^ ~ ^* 8. {^x^-\-^xy + 2xz''^l)dx + dy^-2zdz = 0, 9. (2aj + 2/^ t ^^^) ^^ + 2^2/ dy— dw + o^ dz = 0. 10. Is tlie equation (1 + 2m) xdx + y{l — x)dy + zdz — derivable from a single primitive of the form ^ {x, y, z) =01 11. Shew that any system of lines described on the surface of the sphere x^ -{■ y"^ + z'^ = r^, and satisfying the above equa- tion, would be projected on the plane xy in parabolas. 12. Shew that Monge's method would, if we integrate first with respect to x and z, present the solution of the equa- tion of Ex. 10, in the form {1^2in)x' + z' = j>{y\ 2y{l-x)=-^\y). 13. Applying this form to the problem of Ex. 11, form and solve the differential equation for the determination of (j) (y), and shew that it leads to the result stated in that Ex- ample. 14. Find the equation of the projections of the same system of curves on the plane yz, 19—2 . ( 292 ) [CH. XIII, CHAPTEH XIII. ^f^"^^" SIMULTANEOUS DIFFERENTIAL EQUATIONS. 1. We have hitherto considered only single differential equations. We have now to treat of systems of differential equations. Of such by far the most important class is that in which one of the variables is independent and the others are depend- ent upon it, the number of equations in the system being equal to the number of dependent variables. Thus in the chief problem of physical astronomy — the problem of the motion of a system of material bodies abandoned to their mutual attractions — there is but one independent variable, the time ; the dependent variables are the co-ordinates, which, varying with the time, determine the varying positions of the several members of the material system ; while, lastly, the number of equations being equal to the number of co-ordinates involved, the dependence of the latter upon the time is made determinate. Such a system of equations may properly be called a deter- minate system. We propose in this Chapter to treat only of systems of equations of the above class. And in the first instance we shall speak of simultaneous differential equations of the first order and degree, beginning with particular examples, and proceeding to the consideration of their general theory. Particular Illustrations. 2. The simplest class of examples is that in which the equations of the given system are separately integrable. Ex. 1. Given Idx + mdy + ndz = 0, xdx + ydy -\- zdz = 0, Integrating separately, we have Ix, 4- my -f- nz =c, of + y"^ + z^ = c; and these equations expressing the complete solution of the given system may be said to constitute the primitive system. ART. 3.] PAETICULAR ILLUSTRATIONS. 293 , Another class of examples is that in which, while the equa- tions of the given system are not all separately integrable, they admit of being so combined as to produce an equivalent system of equations which are separately integrable. ^ ^ ^. dx 2x ^ dy 2x Here the first equation alone is separately integrable, and gives ^ = 3 + ? W- Also by addition of the given equations, we have dx + dy dt x + y) therefore '■ — — = dt, x + y log{x+y) = t + c' (h). The primitive system is therefore expressed by (a) and (h). In both the above examples we see that the number of equations of the solution is equal to that of the equations of the system given, and that each equation of the solution in- volves a distinct arbitrary constant. And it is evident that this must be the case whenever we can combine the given equations into an equivalent system of integrable equations of the first order. But as we have not proved that such combi- nation is possible, the following question becomes important, viz. what is the nature of the solution of a system of simulta- neous equations of the first order and degree ? This question will be considered in the next section. General theory of simidtaneous equations of the first order and degree, 8. We shall seek first to establish the general theory of a system composed of two equations between three variables, and therefore of the form Pdx + Qdy + Bdz = 0,] Fdx+ Q'dy+B'dz = 0,] ^^' 29 4f SBIULTANEOUS DIFFERENTIAL EQUATIONS. [CH. XIII. the coefficients P, P', &c. being functions of tlie variable^, or constants. "VVe design to consider tbe above system first, and with the greater care, because there is scarcely any part of the general theory which it does not serve to exemplify. Peop. The solution of the system (1) can always he made to depend upon that of an ordinary differential equation of the second order between two of the primitive variables, and it always consistsoftwo equationsinvolvingtwo arbitrary constants. By algebraic solution of the system (1) we have , RP'-PR' , _ P Q - QP' ^ ,^. As the coefficients of dx in the second members of these equations are functions of x, y, z we may express the reduced system in the form dy = ^ {x, y, z) dx, dz = '^ (x, y, z) dx, whence, regarding x as independent variable, 3| = s^(^>y. ^) (3)> dz -^ = ^^{x,y,z) (4). . Thus the given system enables us to express -^ and -^- by known functions of x, y, z. Now differentiating (3), still on the assumption that x is the independent variable and representing for brevity y -?■ and -vlr, which are known functions of x. ?/, dx dy' dz ^' ' '^' and z. Hence eliminating z bj means of (3) we have a final equation involving -j- , -~ , x, and y. The complete primi- tive of this differential equation of the second order will enable us to express y as a function of x and two arbitrary constants. Suppose the value thus obtained for y to be 3/ = %feCi. cj (6). Then we have by virtue of (3) <^ (^, y, ,) = M^ll^) (7). These two equations involving two arbitrary constants con- tain the complete solution of the system given. 4. It is important to observe that the system (2) may be expressed in the symmetrical form dx dy _ dz QR'-BQ' ~ BF-PE ~FQ' - QF ' If we represent the denominators of the above reduced system by X, F, Z, it becomes dx dy dz /o\ x:t=^-- • («)• This, then, may be regarded as the symmetrical form of a system composed of two differential equations of the first order. Again, the complete solution of such a system, as is expressed by (6) and (7), consists of two equations connecting the varia- bles x, y, z with two arbitrary constants. If we solve these equations with respect to the constants, the solution assumes the form ^{x,y,z)=c^, ^^{x,y,z)^c^ (9). Thus a system of two differential equations of the first order may, without loss of generality, be presented in the symmetrical form (8), and its complete solution in the sym- metrical form (9). 296 SIMULTANEOUS DIFFERENTIAL EQUATIONS. [CH. XIII. Ex. 1. Given (51/ + 9^;) dx ■{■ dy + dz = 0, (4y + 3^) dx + 2dy —djs = 0. Here we find by algebraic solution l=-^^-4- («)' 1=-%-^^ ®' ^Ybence -7^2 = - 3 -/ - 4 -y- aa; dx dx = -3^ + 8y+20^,by(6). Eliminating s by (a), we have on reduction dx^^^Tx^^y-^^ a linear equation with constant coefficients whose complete primitive is y ^ C.e-' + Cf"" (c). d,ii Equating tiie value oi--~ hence determined with that given in {a) we have 3z/ + 4^ = a,6-^+7C4€-^" {d). The complete solution is therefore expressed by (c) and {d). Theoretically it is of no consequence which of the primitive variables we assume as independent. But practically the question is of some importance as affecting the character of the final differential equation. Ex.2. Given^~3aj + 2/==0, ^^-x-y^O. Differentiating the first equation we have d^x ^dx ^y _r^ lif^ ~di'^~di~ ' du from which eliminating -^ by the second equation we have Cl X f^ OjX _ ^-3^ + x + 2/ = 0. AET. 4.] PARTICULAR ILLUSTRATIONS. 297 Hence eliminating 3/ by the first equation Integrating and this value of x substituted in the first equation gives The last two equations constitute the primitive system. We choose next an example in which the given system in- volves functions of the independent variable in the second members. Ex.3. Given ^+ 5a;- 23/ = e', ^-ic + 62/ = €^ Here, differentiating the first equation, we have ^x ^dx s}^y _ t df'^^di^t'^^ Eliminating -f^J the second equation of the given system, we have And, eliminating y by means of the first equation of the system, ^+ll^+28a; = 7e* + 2e% a linear differential equation of the second order whose solu- tion is Hence, by the first of the given equations, 22/ - 5;r + e' = - 4C^e-'^ - 7C^e-'' + ^ e'+ ^ e'\ The last two equations are the complete primitives of the system given. 298 SIMULTANEOUS DIFFERENTIAL EQUATIONS. [CH. XIII. 5. The above theory may be extended to all systems which are composed of n differential equations of the first order and degree connecting ^ + .1 variables. Assume x (independent) and x^, x^, ...x^ (dependent) as the variables of the sj^stem. Then there exist 7i differential equations of the form Fdx + P^dx^ + P/x^.,. + PJx^ = (10), P, P^, &c. being functions of the variables. These equations exactly suffice to determine the ratios of the differentials dx, dxj^,,.. dx^, and thus assume the symmetrical form iJjJL/ LLJU^ Lltl/^y CtvO *^ ^ » x = z;=x,-=^ ■■^^^'' X, Xj, '&c. being determinate functions of the variables. This premised, the solution of the system (11) depends upon the solution of a single differential equation of the n^^ order connecting two of the variable^. Let us select for the two a? and a?,. ^ ,(12). (11) gives < dx^ _ X, dx X * dx~ X' ' ti n ••• dx" X Differentiate the first of these n — 1 times in succession, re- garding X as independent variable and continually substitut- ing for -j^ , ... ~~ their values as given by the w — 1 last equations of the above system. We thus obtain, including the equation operated upon, n equations connecting dx^ dx^ '" dx"" witb the primitive variables and therefore enabling us, 1st, to express the above n differential coefficients in terms, of those variables, 2ndly, by elimination of the n — \ variables, x^, x^, ,..x^, to deduce a single equation of the form ART. 5.] PAETICULAR ILLUSTRATIONS. 299 Now this being a differential equation of the n^^ order, there exist, Chap. IX. Art. 1, n first integrals involving n distinct arbitrary constants and capable of expression in the form F. dx, di^x, d' X, ''''^^' dx' dx' I JLy JU^ F^ [x, X., -^ dx ' d\ dx^ dx''-' d'^-'x. dx" = a a .(14). ^» r ""'' 'dx Gj X^ Cv x^ = c If in this system we substitute for -—^ , -^rr • • • ? «-i their •^ dx ' dx^ dx"" ^ values in terms of the primitive variables above referred to, we shall obtain a system of 7i equations of the form (j)^{x,x^,x^...xj = C^^ \ ^ a-\-maJ ART. 7.] WITH CONSTANT COEFFICIENTS. SOI provided that we determine m so as to satisfy the condition m— ,, a + ma or aV+(a-6')m-6 = (h). Now (a) gives dec -\- mdy , ,. , , ^ — 7 = (a -f ■ mu) dty G + me ^ a + ma whence on integration log (^ + ^y + ^^^^> ) = (^ + ma) t + C (c). In this equation it only remains to substitute in succession the two values of m furnished by {h). The two resulting equations, in which the arbitrary constants must of course be supposed different, will express the complete solution of the problem. When the values of m are equal, the form (c) furnishes directly only a single equation of the complete solution. We may deduce the other equation, either by the method of limits (assuming the law of continuity), or by eliminating x from the given system by means of (c), and then forming a new differentiar equation between 7/ and t. It seems preferable however to employ the general method of Art. 5, by which all difficulties connected with the presence of equal or imagi- nary roots are referred to the corresponding cases of ordinary differential equations. 7. Simultaneous equations are so often presented under the symmetrical form (11) that the appropriate mode of treatment deserves to be carefully studied, especially as it possesses the superiority, always in point of elegance, and frequently in point of convenience, over other processes. It is known that each member of a system of equal frac- tions is equal to the fraction which would be formed by 302 LINEAR EQUATIONS OF FIEST ORDER [CH. XIII. dividing any linear homogeneous function of their nume- rators by the same function of their denominators. Hence if "we have a system of equations of the form dx^ _ dx^ _ dx^ _ dt X~X ~"T~T (16), in which we suppose i the independent variable, and T a function of t only, then we shall have dt _ dx^ + mdx^ . . . + rdx^ ,, ^_» T~ X, + mX^... + rX^ ^^^^V Hence, should the first member be an exact differential, the inquiry is suggested whether the multipliers m, ... r cannot be so determined, whether as functions of the variables or as constants, as to render the second member such also. Now when the system of equations is linear and with constant co- efficients this can always be effected. It may be observed that the character of the system is as manifest from inspec- tion of the symmetrical form (16) as of the ordinary form. If the system be linear and with constant coefficients the de- nominators X^, X^, ... X^ will, when considered with respect to the dependent variables ^^, x^, ,,, x^y be linear and with constant coefficients. In the employment of this method it is often of great ad- vantage to introduce a new independent variable, and to consider all the variables of the given system as dependent upon it. We are thus enabled to secure the condition above adverted to, of having one member of the symmetrical system an exact differential. Ex.l. Given '^'^ ~ '^^ ax + by+ c ax + b'y + d Let us introduce a new variable t so as to give to the system the form dx dy dt , » ax-\-hy ^-c ax -f- h'y + c t ART. 7.] WITH CONSTANT COEFFICIENTS. 803 Here the third member being an exact differential, we shall write dt dx-\-mdy ' t dx + hy + c + m {ax + b'7/ + c) dx + mdi/ (a + ma) x+(b-i- mb')y + c + mc _ 1 (a + ma) dx-\~ {a-\- ma) mdy a + mal {a + ma) x-\-{b-\- mbl) y-\-c-\- mc ' The second member of this equation will be an exact differ- ential if we have {a + ma') m = h + mh' {h), the integral corresponding to each value of m thus deter- mined being of the form 1 ho;t+ C= — , loQ[ f (a + ma) x + (b + ml/) v+c + mc}, ° a + ma ^ or C't = {ax + hy + c + m {ax + Vy + c)] «+'''* . If the roots of the quadratic (6) are m^ and mg, we thus find Cf = [ax -Vly + c ■\- m^ {ax + I'y + c')}^+^^ I , ^ C^ ={ax+hy -{-c + m^ {ax + Vy + c')}»+'^2» J for the primitive equations of the system (a). Those of the given system will be obtained by eliminating t. The result assumes the remarkable form \ax + hy + c + mj^ (ax + Vy + c')}^+"^i* 1 =^ W- {ax + hy + c + m2 (ax + Vy + c')}«+"^20i' ^ _ ^. dx dy dz , xLiX. 2. Given -v? = -^ = -^ , wnere X = ax -\- hy -{- cz ■\- d Y = ax + Vy + cz + d'\ (a). Z — a'x + 6"2/ + c z + c?' 804 LINEAR EQUATIONS OF FIRST ORDER [CH. XIII. Introducing a new variable t, so as to give to the system tlie more complete form dt _dx _dy _dz T~X"T~Z .(^), we have dt _ Idx 4- mdy + ndz 1 V (d). IX+mY+nZ ^ Idx + mdy + ndx . . \ (Zic + m2/ + nz-\-r) ^ '" Provided that we assume al + am + d'n = \l ' hi + h'm + Vn = Xm cl + cm + c'n = \n dl + d'm + d"n = Xr ^ The first three of these may be written in the form {a-'X)l+ am + a"n = O^j U-\-{h'-\)m-\-h'n = (y> (e), cl + cm, + (c" — X) w = Oj whence eliminating I, m, n we have the well-known cubic (a-X) (6'-X) (c"-X)-rc' (a-\) - ca' {h' - X) - la! {c" - X) + a'6"c + a he' = 0. . . (/). Now let the values of X hence found be X^, Xg, Xg, and the corresponding values of I, m, n, r be \, m^, n^, r^, ^2' ^2' ^^'y then integrating (c) we shall have the system _i cf = {l^x + ??2j2/ + n^z + rj^s C2* = (?2^ + ^^22/ + ^2^ + ^2)^% 1 cj, = {l^X + W232/ + 7?3^ + ^3)^'. Hence eliminating t by equating its values, we find as the general solution of the original system of equations ART. 8.] WITH CONSTANT COEFFICIENTS. 305 1 i (l^x + m^y + n^z + r J^i = G (l^x + m^^/ + n^z + r J^^ = C {l,x+m^y + TZg^^ + 7-3)^3. . . (^). In the same way we may integrate the general system 1 2 n X^ X^ " XJ where X^,X^,...X^ are any linear functions of the variables. 8. From the above results the solutions of various sym- metrical systems in which the denominators are not linear may be deduced. The most remarkable of such deductions is the following. Suppose that in the system dx _ dy dz ax +by -\-cz ax-\-oy+cz ax+oy+cz ^^ the solution of which is known from what precedes, we sub- stitute x' = xz, y' = yz, X and y being new variables introduced in the place of x and y . The result is zdx 4- xdz _ z'dy + ydz dz ax + hy ■\-c~ ax+ h'y + c~ a'x + b"y + c' ' to which we may obviously give the form z'dx _ z'dy ax + by + G—x {a'x + b"y + c") a'x +b'y+c— y(a'x + b"y + c") _ dz a x+ Q y -\-c Dividing the first equation of this system by z, we have dx dy ax-\-by+c—x{a'x-[-b"y-\-c') ax-{-b'y-\-c—y{a"x-\'b"y-\-c")" Now this on clearing of fractions will be found to be of the same form as Jacobi's equation (Crelle, Tom. xxiv. p. 1), whose solution on other grounds has been explained, Chap. v. Art. 8. B. D. E. 20 806 LINEAE EQUATIONS OF FIEST OEDER [CH. XIII. We see tliat tlie solution of {h) is deducible from that of the system (a) by changing a/ into xz, y into y^, and elimi- nating z. And just in this way the solution of any sjnumetrical non-linear system of the form X, — x^X X.y — x^X '" X„ — xX 11 2 2 71 71 in which X, X^, X^,...X^ are linear functions of the variables x^, x^,.,.x^ may be made to flow from that of a symmetrical system of the form CbX^ lXiX~ CL'i^ X^T'-'^X ii_r:ri2 _r::r-hi (19)^ in which X^, X^,,..X^^j^ are linear homogeneous functions of the variables x^, x^,...x^_^^. The general solution of the sys- tem (18) seems to have been first obtained by Hesse {Crelle, Tom. XXV. p. 171). 9. Lastly, certain systems of linear equations which have not constant coefficients may be solved by the above method. Thus the solution of the equations ^^ + T{ax + hy) = T, %^T[dx + yy) = T, (a), where T, T^, T^ are functions of the independent variable, may be reduced to that of an ordinary linear differential equa- tion of the first order. For proceeding as before, we find ^^^±^ + XT(^ + m2/)=T, + m2; (5), provided that X and m be determined by the conditions X = a + ma J \m = h + mh' (c). AET. 10.] WITH CONSTANT COEFFICIENTS. 307 Hence eliminating X, we have m (a + ma) = h + mb' {d)j which gives two values for m. Integrating (b) regarded as a linear equation of the first order between x + my and t, and substituting for X its value in terms of in given by the first equation of the system (c), we have in which it remains to substitute for m its values given by {d). Ex. Given ^ + ^ (^-2/) = 1, -l + l{x+^y)^t The solution is If in the system (a) we make T=l, it becomes a system of equations with constant coefficients but possessed of second members. The general system analogous to (a) when the number of variables is increased, may be solved by the same method. It may be well to notice that the equivalent symmetrical form is where X^, X^,...X^ are linear homogeneous functions of the dependent variables, and T, T^,...T^ are functions of t Treated under this form, it is obvious that its solution will be made to depend upon that of a linear differential equation of the first order, and an auxiliary algebraic equation of the n^^ degree. Equations of an order higher than the first. 10. Any system of simultaneous equations of an order higher than the first is reducible to a system of the first 20—2 308 EQUATIONS OF AN OEDER [CH. XIIT. order. And this reduction though not always necessary for the purpose of solution is theoretically important, because it enables us to predicate what kind of solution is possible. To effect this reduction it is only necessary to regard as a new variable and to express as such by a new symbol, each differential coefficient, except the highest, of each dependent variable in the given equations. The transformed equations will thus be of the first order, and the connecting relations of the first order also ; and the two together will constitute a system of simultaneous equations of the first order. Ex. Given the dynamical system ~df~ ' ■^-^' df~ ' where X, Y, Z are functions of the variables. Here if we assume dx _ , dif _ , dz _ , Tt~^' tt~^' dt^^' the given system assumes the form (^^ XT y v" ^^^ 'z 'di~ ' dt~ ' ~dt~ Thus we have in the whole six equations of the first order between the six dependent variables x, y, z, x\ y', z, and the independent variable t. The complete solution of the latter system will therefore consist of six equations connecting the above system of varia- bles with six arbitrary constants. If from these six equations we eliminate the three new variables x', y, z\ we obtain three equations connecting the original variables x, y, 2, t with the above-mentioned six arbitrary constants. And thus it might be shewn that the complete solution of any system of three differential equations of the second order between four variables will be expressed by three primitive equations connecting these variables with six arbitrary con- stants. ART. 10.] HIGHER THAN THE FIRST. 309 And still more generally, the complete solution of a system of n differential equations containing n + \ variables of which one is independent will consist of 7i equations connecting those variables with a nuwher of constants equal to the sum, of the indices of order of the several highest differential coefficients. For let t be the independent and x one of the dependent variables, and let the highest differential coefficient of x d^x which presents itself be -7^ . Then in the reduction of the system of given equations to a system of equations of the first order it is necessary to introduce n — \ new variables con- nected with X by the relations ax (X/X^ ax _^j dt ^' dt "^'••* dt ~ "-^* Thus the number of variables in the transformed system cor- responding to X and its differential coefficients will be n, and as a similar remark applies to all the other variables, it ap- pears that the total number of variables of the transformed system will be equal to the sum of the indices of the orders of the highest differential coefficients of the several dependent variables in the system given. Such then will be the number of equations of the transformed system, and such the number of constants introduced by their complete integration. Art. 5. It is also evident "that if from the equations by which the complete solution is expressed we eliminate all the new variables there will remain a number of equations equal in number to the original equations, and connecting the primi- tive variables with the constants above mentioned. Thus the proposition is established. The transformation above employed is further important, because in the highest class of researches on theoretical dy- namics it is always supposed that the differential equations of motion are reduced to a system of simultaneous equations of the first order. At the same time it is not necessary for ordinary purposes to effect this reduction. Differentiation and elimination al- ways enable us to arrive at a differential equation, higher in order, between two of the variables. The method of indeter- 310 EQUATIONS OF AN OKBER [CH. XIII. minate multipliers may also be sometimes used witli advan- tage. No general rule can however be given. [The statement respecting the number of arbitrary constants is not universally true. Suppose, for example, that there are two simultaneous differential equations which connect x and y with the independent variable t. Let one equation contain differential coefficients up to -, and -7^ inclusive ; and let the other equation contain differential coefficients up to -yy and -^ inclusive : then it can be shewn that the number of arbitrary constants involved in the solution is the greater of the two numbers m-\-s and n-\-r. See Cournot, Traite EU- mentaire de la Theorie des Fonctions...l84il. Vol. 11. p. 318.] d^£G d^v Ex. 1. Given -7-7 = ax + hy, -A = dx + Vy, 1st method. Differentiating the first equation twice with respect to ty we have d^x __ ^x , ^y dt'~''de'^^~de' d^V Eliminating y and -^ from the above three equations, we have djX d X ^ -{a-\-h')-^+{ah'-dh)x = (a). The complete integral of this linear equation with constant coefficients will determine x, whence y is given by the formula 1 fd'^x \ 2nd method. From the given equations we find d^x dj^y Tp + ^ -7I = (^ + w^a') £c + (& + 'mh') y df ' df = (a + m(i ) a? + — , , y ART. 10.] HIGHER THAN THE FIRST. 311 Let X + my — m, then provided that we determine m by the condition h + mh' ,, X m = , io), a + ma ^ ^ we shall have whence u = (7^6^^+'^^'^*^+ O^e"^^^'"*')*^. Let m^, m^ be the values of m given by (6), then the complete primitive system is and this is really equivalent to the previous solution, though more symmetrical. Ex. 2. The approximate equations for the horizontal mo- tion of a pendulum when the influence of the earth's rotation is taken into account* are j2 , y W, I representing the length of the pendulum, g the force of gravity, and — r being equal to the product of the earth's angular velocity into the sine of the latitude of the place. As the equations have constant coefficients they admit of complete integration. If we differentiate so as to enable us dii (jjU to eliminate ^, -# and -~ , we find as the result g+K^^'-^flS-^?-^ ^^>' * Jullien, Problemes de Mecanique Rationnelle^ Tom. ii. p. 233» 812 EQUATIONS OF AN OEDER [CH. XIII. the complete solution of which is of the form x = Acos {mf -\-a)-\-B cos (m^i + yS) {c), where A, a, 5, j3 are arbitrary constants, and m^, m^ are the two roots, with signs changed, of the equation ^^_2(^2r=' + |)/. + |' = 0. From the above value of x that of y may be obtained by means of the formula which is readily deduced from the given equations. The above system may also be solved by assuming (e). The transformed equations are x = X cos ri + y sin rt y — — x sin rt + y cos rt lere leni de ce we find X^ = r^ + i; X — A cos \t+B sin \t y' =■ A' cos \t -I- B sin \t (/)• 11. In problems connected with central forces particular forms of the following system of equations present them- selves, viz. d^x _ dU d^y _ dR d^z _dR , . W~dx' W'^dy' df~'dz ^^^' where R is a given function of the quantity Aj(x^ + y^ + z^) AET. 11.] HIGHER THAN THE FIRST. 813 or r. Multiplying the above equations by dx, cly, dz respec- tively, and integrating, we have MS)-©"-(S)l=--- «• B being an arbitrary constant. . . . dR dRdr xdR „ ,, . „ Again, since -y- = -^ -- = - — - &c. the ffiven system of ax ar ax r dr ° -^ equations may be expressed in the form d^x _ X dR d^y _ y dR d^z _ z dR df r dr ' df r dr ' df r dr ' dR Now if from each pair of equations we eliminate -7- , we dr obtain d^y d^x _ d^z d^y _ d^x d^z ^ of which it is evident that two only are independent. Inte- grating these, we have dy dx ^ dz dij ^ dx dz ^ dt '^ dt ^ ^ dt dt ^ dt dt ^ Cj, Cg, C3 being constants. Squaring the last three equations and adding, we obtain a result which may be expressed in the form / 2^ 2^ 2N fM^A' , /%\' , (dz\^) ( dx ^ dy ^ dz\^ or, by virtue of (6) and of the known value of r, 2r^(i? + i?)-(rJJ = ^= (c), rdr /* rdr 314 EQUATIONS OF AN ORDER [CH. XIII. Again, it is evident that by means of (c) we can eliminate R from eacli equation of the system (a-). For (c) gives 1 (A' /drV) Substituting which in the first of the given equations, we have (Px _x f A^ dr d dr\ df~r\V'^didrdi) A' d'r^ _xf__A^ dS\ ~r\ r^'^dfj' d^x d\ A^x ^ Hence ^__«;_ + _=0, d ^ d X A^x „ at dt r r therefore r' j^ r' j^ (fj + ^' ^ = 0. Adt If we now assume — ^ = dcj), the above becomes whence - = a^ cos ^ + &j sin (/> (/). In like manner, we find ^=a^cos(l) + h^8m(j) {g), z - = a3C0S(/) + JgSinc^ (/i), in which we must substitute for ^ its value, viz. ,_[Adt_[ Adr ,^ ART. 12.] HIGHER THAN THE FIRST. 315 To this expression it would be superfluous to annex an arbi- trary constant before that substitution. For each of the second members of (/), {g), (h) is expressible in the form (7 cos ((/) + C), in which ^ is already provided with an arbi- trary constant. The solution is therefore expressed by means of [e) and (^), which determine r and the auxiliary as functions of t, and by (/), [g), [h)y which then enable us to express x, y, z as functions of t. As we have however made no attempt to preserve independence in the series of results, the constants will not be independent. If we add the squares of [f], (^), ih), ^^^ shall have 1 = {a^ + a/ -f a^) cos^ ^ + 2 {afi^ + a}>^ + afi^ sin (^ cos <^ + (2>i' + ^/ + ^/)sin>, which involves the relations among the constants The six constants in (/), {g), (h), thus limited, supply the place of only three arbitrary constants, and there being three also involved in (e), the total number is six, as it ought to be. In the same way we may integrate the more general system (^00^ _ dR d^x^ _ dU d^x^ _ dR where i2 is a function of ^J{x^ -^-x^ ,..-\-x^). The results, which have no application in our astronomy, are of the form which the above analysis would suggest. Binet, to whom the method is due, has applied it to the problem of elliptic motion. (Liouville, Tom. ii. p. 457.) For all practical ends the employment of polar co-ordinates, as explained in treatises on dynamics, is to be preferred, 12. The following example presents itself in a discussion by M. Liouville*, of a very interesting case of the problem of three bodies. * Sur un cas partieulier du Proileme des trois corps. Journal de Blathe- matiques, Tom. i, 2nd series, p. 248. 316 EXERCISES. [CH. XIII. Ex. Given -y-^ -\-n^ [u— Sx' {ux + ?;y)} = 0, __ + n^ \y _ 3y {^ux + vy')] = ; where, for brevity, x is put for cos [at + h)y y for sin [at + h). If we transform the above equation by assuming ux' + vy = Uj uy — vx = Yj we find, after all reductions are effected, (J^TT dV And these equations being linear and with constant co- efficients, may be integrated by the process of the previ- ous section. EXEECISES. 1. ^+4^ + 1 = 0, ^+8j/-a? = 0. dt 4i dt ^ dx _ ^y _ J4. 2^ — DX -he* 33 — 0^ + e ' , , CH. XIII.] EXERCISES. 317 6. -dx=Jy-^= ^^ 8. J-S^-4^ + 3 = a g + ^ + 2/ + 5 = 0. 9. -^-ji+m^^ = 0, -^f-m=^=0. 10. Given -^^-^ = ^ = ^^ = ^ , where X= ax + hy + cz, Y = ax + &'?^ 4- c'zj and T, J'j, T^, T^ are functions of t 11. What is the general form of the solution of a system of n simultaneous equations of the first order between n + 1 variables ? 12. What number of constants will be involved in the solution of a system of three simultaneous equations of the first, second and fourth order respectively between four variables ? 13. Of the system of dynamical equations, de ^ t'~ ' df "^ 7-^ ' df "^ r'~ ~ ^' where r = {x^ -{-y^ -^-z^Y, seven first integrals are obtained of which it is subsequently found that five only are independent. How many final integrals can hence be deduced without pro- ceeding to another integration ? 818 EXERCISES. [CH. XIII. 14. Given a '-r- = (h — c)yz (1), h-^ = {c-a)zx (2), c^^{a-h)wy (8). Putting - — =1, =m, :=n we find, on eliminating cZf, — C — CL d "^ Ixdx = mydy = nzdZf from which y and z will be found in terms of x, and their values will reduce (1) to a differential equation of the first order betwoen x and t. Or multiply the given equations, first by x, y, z, respectively, add the results and integrate; 2ndly by ax, by, cz, respectively, add the results and integrate. Then by means of the integrals obtained eliminate two of the variables from any of the given equations. 15. Shew that in the example of Art. 12, the transform- ation x = x cos (rt + e) + 2/' sin (rt + e), y — — X sin {rt •\-i)-\- y cos {rt + e), 6 being an arbitrary constant, would not lead to a more general solution than the one actually arrived at. ( 319 ) CHAPTER XIV. ^ (}v; OF PAETIAL DIFFERENTIAL EQUATIONS. 1. Partial differential equations are distinguished by the fact that they involve partial differential coefficients in their expression, and therefore indicate the existence of more than one independent variable. Chap. i. Art. 2. The nature of these equations will be best explained by one or two examples of the mode of their formation. Ex. 1. The general equation of cylindrical surfaces is OS — Iz = whence d^z ^2= (u), '\jr (v), where u and v are given functions of x, y, z. Then representing the first member by F, regarding a? and y as independent variables, and forming all possible derived equations up to the second order, we have dx ^ dy ' daf ' dxdy ' dy^ which with the given equation make six equations. But these containing the six functions {u), tW, '{u), f'{v), f («), t"W> B. D. E. 21 322 PAETIAL DIFFERENTIAL EQUATIONS. [CH. XIV. do not, in general, suffice to enable ns by the elimination of the latter, to form a partial differential equation of the second order free from arbitrary functions. "We see then, 1st, that partial differential equations do not arise from the elimination of arbitrary functions only ; 2ndly, that even as respects this mode of genesis, no general canons exist similar to those which govern the connexion of ordinary differential equations with their primitives. On both these grounds it will be proper, in considering special classes of equations, to examine their special origin and to seek therein the clue to their solution. Solution of partial differential equations. 2. Before proceeding to general theories of the solution of partial differential equations, it may be noticed that there are some equations of which the solution may be directly reduced to that of ordinary differential equations. This is the case when the partial differential coefficients have ail been formed with respect to one only of the variables. We can then integrate as if this were in fact the only inde- pendent variable, provided that we finally introduce arbitrary functions of the other independent variables in the place of arbitrary constants. Ex.1. Given X -\- y -Y- = 0. ^ ax ■ Multiplying by dx, integrating with respect to x, and adding an arbitrary function of y^ we have the solution required. It is permitted in the above, and in all similar cases, to complete the solution by adding an arbitrary function of y, because, with reference to the integration effected, y is con- stant ; and it is necessary to add such a complementary func- tion in order to obtain the most general solution, because an arbitrary function of one of the variables is more general than an arbitrary constant not involving that variable. ART. 2.] SOLUTION OF PAETIAL DIFFERENTIAL EQUATIONS. 823 Ex. 2. Given y-^-2x-'2^z—y = 0.. This equation may be expressed in the form dz 2 ^ 2x dy y y Involving no differential coefficient with respect to x, it may be treated as a linear differential equation of the first order in which y is the independent, and z the dependent variable ; only instead of an arbitrary constant we must add an arbi- trary function of x. The final solution is X +y-\- z =y^(l) (x). It sometimes happens that equations not belonging to the above class are reducible to it by a transformation. d'z Ex. 3. Given -^-^r- = cc^ + y^ axay dz dw Let -J- = w, then we have —-=03^ + y'^^ whence integrating with respect to y, and adding an arbitrary function of x, dz . . . Restoring to w its value -p , integrating with respect to x, and adding an arbitrary function of y, we have j^{x)dx^^\r{y). _x^y y^x Now <^(x) being arbitrary, \(^{x) dx is also arbitrary, and may be represented by p^ {x), whence X ij "4~ 11 X [See the Supplementary Volume, Chapter xxiv. Art. 1.] 21—2 324 LINEAR PARTIAL DIFFERENTIAL EQUATIONS [CH. XIV. Linear partial differential equations of the first order. 8. When there are but three variables, z dependent, x and y independent, the equations to be considered assume the form yj aZ ^ aZ -j-y dx dy ' P, Q, and R being given functions of x, y, z, or constant. Tiiis form we shall first consider. Usually the differential coefficients -r- and -y- are repre- "^ dx dy sented by j9 and q respectively. The equation thus becomes Fp + Qq = B (1). The mode of solution is due to Lagrange, and was first established by the following considerations. Since ^ is a function of cc and y, we have dz = pdx + qdy. Hence eliminating p between the above and the given equa- tion, we have Pdz — Bdx = q {Pdy — Qdx), Suppose in the first place that Pdz — Rdx is the exact differ- ential of a function u, and Pdy — Qdx the exact differential of a function v, then we have du = qdv. Now the first member being an exact differential, the second must also be such. This requires that q should be a function of V, but does not limit the form of the function. Represent it by (ji\v), then we have du = (j)'(v) dv, whence u=4>{v) (2). The functions u and v are determined by integrating the equations ■ Pdz-Bdx = 0, Pdy-Qdx^^O, ART. 3.] OF THE FIEST OEDEE. 825 symmetrically expressible in the form dx _dy _dz . ~F^'Q~~R ^^^' and of whicli tlie solution, Chap. xiii. Art. 5, assumes the form u = a^ v=h (4), a and h being arbitrary constants. Dismissing the particular hypothesis above employed, La- grange then proves that if in any case we can obtain two integrals of the system (3) in the forms (4), then u — (f>{v) will satisfy the partial differential equation, in perfect indepen- dence of the form of the function ^. We shall adopt a somewhat different course. We shall first establish a general Rule for the formation of a partial differential equation whose primitive is of the form ti = (f> (v), u and V being given functions of x, y, and z. Upon the solu- tion of this direct problem we shall ground the solution of the inverse problem of ascending from the partial differential equation to its primitive. Peoposition. a primitive equation of the form u = <^ [v), where u and v are given functions of x, y, Zy gives rise to a partial differential equation of the form Pp+Qq = B (5), where P, Q, R are functions of x, y, z. Before demonstrating this proposition we stop to observe that the form u = (^ (y) is equivalent to the form f(u,v) = 0, f(u, v) denoting an arbitrary function of u and v. For solving the latter equation we have u= (f> {v). It is also equivalent to F{x,y,z,cj>(iv)]=0, (j) being an arbitrary, but F a definite functional symbol. 326 LINEAR PARTIAL DIFFERENTIAL EQUATIONS [CH. XIV. For solving the latter equation with respect to ^ (v) we have a result of the form (j) {v) = F^ {x, y, z), or (f){v) = u on representing F^^ (x, y, z) by u. Thus the proposition affirmed amounts to this, viz. that any equation between x, y, and z which involves an arbitrary function will give rise to a linear partial differential equation of the first order. Differentiating the primitive u = ^ (v), first with respect to X, secondly with respect to y, we have du du ,, , . fdv dv \ du du ,r , s fdv . dv \ Eliminating cj>' (v) by dividing the second equation by the first, we have du du dv dv dy dz ^ _ dy dz ^ du du dv dv * dx dz-^ dx dz^ or, on clearing of fractions, fdudv dudv\ fdudv dudv\ \dy dz dz dy) ^ \dz dx dx dz) ^ o dudv dudv dxdy dy dx Now this is a partial differential equation of the form (5). For u and v being given functions of x, y and z^ the coefficients of p and q, as well as the second member, are known. The proposition is therefore proved. As an illustration, we have in Ex. 1, Art. 1, u = x—lzy V = y — mz, whence du _^ du _^ du __ , dx~ ^ dy * dz ' dv _ ^ dv _^ dv _ dx ^ dy ^ dz (6). ART. 4] OF THE FIRST ORDER. S27 Substituting these values in (6) there results, Ip + mg^ = 1, which agrees with the result before obtained. 4. The general equation (6), of which the above theorem ^ is a direct consequence, has been established by the direct elimination of the arbitrary function. But the same result may also be established in the following manner, which has the advantage of shewing the real nature of the dependence of the coefficients P, Q, R upon the given functions u and v. [See a Note at the end of the volume.] Differentiating the equation u — <^ {v) with respect to all the variables, we have and as this equation is to hold true independently of the form of the function (f> (y), and therefore of the form of the derived function ^' (v), we must have dii , du J du J ^ -^dx + -T- dill + -7- a^ = ax ay as dv , dn T dv , ^ -^ dx + -r dy ■}■ -J- dz =0 dos dy '^ dz (8),. whence we find dx dy- __ dz dudv dudv dudv dudv dudv dudv dy dz dz dy dz dx dx dz dx dy dy dx ... (9). Introducing now the condition that z is the dependent, X and y the independent variables, we have pdx + qdy = dz. To eliminate the differentials, let the terms of this equation be divided by the respectively equal members of (9), and we have o28 LINEAR PARTIAL DIFFERENTIAL EQUATIONS [CH. XIV. 'dudv dudv\ fdudv dudv\ diidv dudv\ fdudv dudv\ dy dz dz dy) ^ \dz dx dx dzj _ dii dv du dv ,^ „v dxdy dy dx whicli agrees with (6). Now if in the above general form we represent as before the coefficient of ^ by P, that of q by Q, and the second member by R, we see from (9) that P, Q, R are proportional to dx, dy and dz, in the system (8). But that system is precisely the same as we should obtain by differentiating the equations a and h being arbitrary constants. Hence, the partial differ- ential equation whose complete primitive is w = ^ (v), may be formed by the following simple rule. Rule. Forming the equations u = a, v = h, where a and h are arbitrary constants, differentiate them, and determine the ratios ofdx, dy, dz in the form dx _dy _dz .- - . F~'Q'~R ^ ^• TJien will Pp + Qq = Rhe the differential equation required. Or, the Rule may more briefly be stated thus. Eliminate dx, dy, dz between the three equations, du = Oj dv = 0, dz—pdx — qdy=0 (12). It is worth while to notice that the partial differential equa- tion here presents itself, like many other results of analysis, in the form of a determinant. Ex. The functional equation of surfaces of revolution, the axis passing through the origin, is Ix + my -h nz = ^ {x^ + y"^ + /) ; their partial differential equation is required. ART. 5.] OF THE FIEST OEDER. 329 Here, proceeding according to the Rule, we have Idx + mdy + ndz = 0, xdx + ydy 4- zdz = 0, , do) dy dz whence ■ = ^-r- = ^ . mz — ny nx — Iz ly — nix The partial differential equation therefore is {mz — ny) p + (nx — Iz) q = ly— 'nix, [See the Supplementary Volume, Chapter xxiv. Art. 2.] 5. We proceed in the second place to apply the above results to the inverse problem of solution. From what has been said of the origin of partial differential equations of the form Fp + Qq = B it is evident that their solution will be effected by the following rule. E.ULE. Form the system of ordinary differential equations dx dy dz ,^^. T'^Q-B ^^'''' and express their integrals in the forms u = a, v = h; then will the equation u=f (y), where f is a symbol of arbitrary function- ality, express the solution required. For, setting out from the assumed primitive, u=f(v)y we should, by the application of the previous and direct Rule, be led to the partial differential equation in question. The difficulty of the process consisting therefore solely in the integration of the system of ordinary differential equations (18), is referred to the methods of the last Chapter. Ex. 1. Given xp-\-yq= nz. Here, the system of ordinary differential equations is dx _ dy dz X y nz* and the variables therein are separated. The integrals may obviously be expressed in the forms y _ ^ _ p ~~ — %^) n~~ ^ ' X X 830 LINEAR PARTIAL DIFFERENTIAL EQUATIONS [CH. XIV. Hence, the required solution is ^ -A(y indicating that -sr is a homogeneous function of x and y of the 7if^ degree. Ex. % Given {mz — ny) p + (nx — h)q = Iy — mx. Here the system of ordinary differential equations is dx dy __ dz "mz — ny iix — Iz ly — nix ' From these we readily deduce Idx + ondy + ndz = 0, xdx + ydy + zdz = 0, the integrals of which are Ix + my + nz = a, x^ -\-y^ + z"^ = h, the final solution is therefore Ix + my + nz= (f>(x^ + y^ + z^). Ex. 3. Given (fx - 2x') g + (2/ - x'y) ^ = d{x'- y') ^. This is the partial differential equation on the solution of which would depend the determination of the general inte- grating factor of the equation (x^y — 2y^) dx + (y^x — 2^*) dy=0. Chap. IV. Art. 3. The system of ordinary differential equations is dx dy _ d/jL y'x - 2^* ^ 2y^-x'y~ 9 {x^'-y^)fM "" . The first equation of the system is {x^y - 2y') dx + (y'x - 2x') dy = 0, and of this the complete solution is X y ' -^ + — 2 == ^'^ (a). ART. 5.] OF THE FIRST ORDER. 331 We may also deduce from (a) \x yj fM of which the complete primitive is 3 3 r xy [jtj = c . Hence the solution of the partial differential equation is 1 [ X y\ .and this agrees with the result obta^ined by other considera- tions in the Chapter referred to. "We may note that in this, as in all similar cases, the differ- ential equation whose integrating factor is sought, presents itself as one of the equations of the system on whose solution the complete determination of the factor rests. To complete the theory of the linear partial differential equation Pp •\-Qq — R it ought to be shewn that the solu- tion u=f{v)j or as it may be expressed, F{u,v)=0 (14), includes every possible solution. Let X (^} y> ^) — ^j ^^ f^^ simplicity % = 0, represent any particular solution. Differentiating, we have dx dz^ ' dy dz^ ' and substituting the values of p and g^ hence derived in the given equation dx dy dz Similar equations being obtained from the particular in- tegrals u = a,v = h,y^Q have, on eliminating P, Q, R, dx (ciu dv du dv\ dx fdu dv du dv\ dx \dy dz dz dy) dy \dz dx dx dz) dx (du dv du dv\ _^ „ ,^^. dz\dxdy dy dx) ^' 332 LINEAK PARTIAL DIFFERENTIAL EQUATIONS [CH. XIV. Now suppose the forms of u and v to be u=^{x,y,z)y v = f{x, y,z) (16), ^ (x, y, z) and a/t (a?, y^ z) being given functions. From these two equations some two of the quantities x^ y, z may be de- termined as functions of the other and of u and v. Suppose X and y thus determined as functions of z^ u, and v ; then by substitution ^ (x, y, z) becomes a function of z, u, and v, and we may write Hence we- find dx du dx dv dx^ d/x^^dx^dM^dx^dv^ dy du dy dv dy ' dx^dx^du ^dx^dv ^ dy^ ^ dz du dz dv dz dz ' Substituting these in (15) and reducing, we have ^Xi (^^^ ^^ du dv\_ ^ ,^^K dz \dx dy dy dx) But, were the second factor of the first member equal to 0, u would be a definite function of v and z (Chap. Ii. Art. 1) and the equations (16) could not determine x and y as by hypothesis they do. We have then -^ = 0, whence Xi ^^es not involve z. Thus, X being expressible as a function of u and v, the equation ^ = is included in the general form (14). [See the Supplementary Volume, Chapter xxiv. Art. 3.] 6. The above theory may be obviously extended to partial differential equations of the first order and degree involving any number of variables. ART. 6.] OF THE FIRST ORDER, 883 Let ajj, 5?2 ••• ^n represent the independent variables and z the dependent variable. Let moreover the primitive func- tional equation be expressed in the form ^ = ^(^1. ^2---0- (18), where u,v^^v^.,. v^_^ are known functions of the variables. Differentiating with respect to all the variables, and for brevity representing ^ (v^, ^2 ••• '^n-i) ^J ^> ^^^ have du = -T- dv.-{- -r~ dv^... dv, ^ dv„ ^ 'n-1 But being an arbitrary function of the quantities v^, v^ ... v^_^, it is evident that the supposition that the above equation is generally true involves the supposition that the system of equations du = 0, dv^ =0, dv^ — O,... dv^_^ = 0, is true, a system of which the developed form is du , du -, du y ^ -j— da^,...-i- -J— dx^ -^ ~r dz = K)-\ dx^ ""^^ ' * ' ' dx^ ^^"^ ' dz dv, , dv, J dv. , _ j-^ dx..:.-\- -T^dx^-\-—^dz = ax, ^ ax„ ^ dz r -^=^dx,... + -j^^dxn+—^dz = J dx. ^ dx dz 1 n (19). Now this system may be converted into an equivalent sys- tem determining the ratios of the differentials dx^^, dx^... dx^^, dz, in the form UiX^ (ajX„ ax az (20), where P^,P^...P^ and R are functions of the variables or are constants. Introducing the condition that z is to be regarded as a function of a?^, iCg, . . . ^„, we have p^dx^-\-p^dx^...'-VpJx^ = dz (21), 334 LINEAR PARTIAL DIFFERENTIAL EQUATIONS [CH. XIV. where p^, p^-^-p^ ^^^ "the several first differential coefficients of z. And now eliminating the differentials dx^, dx^, . . . dx^, dz from (20) and (21) by division, we have P,i,. + P,i,,... + P.p„ = i? (22), for the partial differential equation sought. Conversely, to integrate the above equation it is only neces- sary to form and to integrate the system (20). Representing the integrals of that system in the forms the final solution will be u = (P(v^, v^,...v^_^ (23).- This solution may also be put in the form ^{u,v,,v^,...v,_^ = (24). Lagrange, Memoir es de I' Academic Boy ale de Berlin, 1779, p. 152. Here the auxiliary system of equations is dx _ dy _ dz __ dt y + z + t z-^x + t x + y -i-t x + y + js' which is reducible to the form dt — dx _dt — dy __dt — dz _dx + dy+ dz -h dt x-t ~ y-t ~ z-t ~ S(x+y + z + t) ' each term being now an. exact differential. The system of integrals will evidently be G, = -^ = (a;+y + z + t)i. x — t y — t z — t Or, representing the function aj + ^z + ^ + iby S, AKT. 7.] OF THE FIRST OKDER. 335 Whence the complete integral symmetrically exhibited will be ^ is"^ {x - 1), S^ {y-t), Si {z - 1)\ = 0. The solution of all partial differential equations of the form ^ dZ -TT- CLZ -y ctz ^ ^ dx^ ^ dx^ '" " dx^ ' where X^, X^,... X^ and Z are any linear functions of the variables x^, x^y...x^^, z, may be completely effected. For it depends on the solution of the system of ordinary differential equations dx, dx„ dx dz Xj X/" X^ Z' which has been fully discussed in Chap. xiii. Hesse has integrated the still more general equation which, according to the above notation, would present itself in the form Y ^—M Y A^ Y ^^ 'dxj^'dx/"'^ ""dx^ ^ f dz , dz dz \ ^ where X^,X^,... X^^^ are any linear functions of the variables. (Crelle, Tom. xxv. p. 171.) [See the Supplementary Volume, Chapter xxiv. Arts. 4. . .7.] Non-linear equations of the first order with three variables. 7. Partial differential equations of the first order with two independent variables w, y, and one dependent variable z, have for their typical form F{x,2/, z,p, q) = (1). Those which are linear with respect to p and q, we have considered apart. Those which are non-linear we proceed to 336 KON-LINEAR EQUATIONS OF THE FIRST [CH. XIV. consider. The genesis of an equation of tliis class from a com- plete primitive involving two arbitrary constants has been illustrated in Ex. 2, Art. 1 ; and the mode is general. From a given primitive, involving x, y, z with two arbitrary con- stants, and from its two derived equations of the first order formed by differentiating wjth respect to a? and y respectively, it is possible to eliminate both the constants. The result is a partial differential equation of the first order. Conversely the integration of such an equation consists mainly in the discovery of its complete primitive — not that this is its only form of solution, but because out of it all other forms may be de- veloped. From the complete primitive involving arbitrary constants arise, 1st, the general primitive involving arbitrary functions; 2ndly, the singular solution. The terminology of Lagrange is here adopted. {Calcul des Fonctions, Legon XX.) To deduce the complete primitive of a partial differential equation of the form F {x, y, z, p, q) = 0. The existence of a primitive relation between x, y, z in- volves the supposition that the equation dz =pdx + qdy (2), should satisfy the condition of integrability, 'dp\ (df dy) \dx) * '" " where [-r] represents the differential coefficient of p with respect to y on the assumption that p is expressed as a func- tion of X and y, and f -^ j the differential coefficient of q with respect to x, on a similar assumption as to the expression of ^. Now regarding p for the sake of greater generality as a function oi x, y, z, z being at the same time an unknown function of x and y, we have dp\ _dp dp dz dy) dy dz dy _dp dp dy ^ dz ' ART. 7.] ORDER WITH THREE VARIABLES. 337 Again, suppose that by means of the given differential equation, q may be expressed as a function of x, y, z, p. Re- garding in such expression s as a function of x, ?/, and ^ as a function of x, y, and z, we have (olq\ _ dq dq dz clq fcfp dp dz\ \dxJ dx dz dx dp \dx dz dxj ' ^dq dq dq dp dq dp dx dz ^ dp dx dp dz Substituting these values in (3), we have on transposition _d^d^^d2^( dq\dj,^^_^ Sq dp ax dy \ dp) dz dx dz Now the coefficients —-r-t ^ — p-p } ^^^^ t^® second member -J -\-'P^ being known functions of x, y, z, p, since q as determined by the given equation is such, the above presents itself as a linear partial differential equation of the first order in which jj is the dependent and x, y, z the independent variables. Applying therefore Lagrange's process. Art. 6, we have the auxiliary system dx _, dz _ dp ,^. ^dq " ^~^~__ 'dq~ d/i dq^ ^^^ ' dp ^ ^ dp dx ^ dz and this, it is to be observed, is a system of ordinary differen- tial equations between x, y, z, and p. It may further be noted that while it has been formed in order to secure the integrability of the equation dz =pdx + qdy, it also includes that equation. For it gives dz = (q — p -r-j dy = pdx + qdy, since by the equation of the first and second members ^-ldy = dx, dp '^ B. D. E. 22 338 NON-LINEAR EQUATIONS OF THE FIRST [CH. XIV. Accordingly if from the system (5) we can deduce a value of p involving an arbitrary constant, that value together with the corresponding value of q drawn from the given equation will render the equation dz =j)dx + qdy integrable. Effecting the integration we shall obtain an equation between x, y, z and two arbitrary constants which will constitute a complete primitive. We say a and not tlie complete primitive, because the sys- tem (5) may furnish more than one value of 'p involving an arbitrary constant, and so give occasion to deduce more than one complete primitive. Lagrange had indeed proposed to employ the general value of ^ involving arbitrary functions, furnished by the solution of the partial differential equation (4). The sufl&ciency of a value involving only an arbitrary constant was remarked by Charpit and subsequently recog- nised by Lagrange. The practical rule for the discovery of a complete primitive of the equation F{x, y, z, p, ^) = is therefore the following. Express q in terms of x, y, z, p. Substitute this value in the auxiliary system (5), and deduce by integration a value ofp involving an arbitrary constant. Substitute that value of p with the corresponding value ofq in the equation dz =pdx + qdy, also included in the auxiliary system (5), and again integrate. Ex. 1. Required a complete primitive of the equation z=pq. Substituting - for q, the system (5) becomes "ifdx , pdz ^ z y + a The equation dj) = dy gives p = y + a, whence q = Therefore dz = (y -\- a) dx -\ dy, of which the integral is z = {y-\-d){x^l) (6), a and b being arbitrary constants. This then is a complete primitive. ART. 8.] ORDER WITH THREE VARIABLES, 889 Another will be found by employing the equation integrating which, we have 1 whence dz=^cz^dx-\ — dy. Integrating, we find or ^z'^ — ex -\- - y ■\- e, ^= IT- -(^^^ e being a new arbitrary constant. It will be found on trial that both (6) and (7) satisfy the equation z =pc[, 8. Prop. Given a complete primitive of a partial differ- ential equation of the first order, to deduce the general primi- tive and the singular solution. Expressing the complete primitive in the form z^fix, y, a,h) (8), a and h being its arbitrary constants, the partial differential equation is itself obtained by eliminating a and b between the above equation and the derived equations _ df(x, y, a, h) _ df(x, y, a, h) ^~ dx ' ^~ dy ' or, as we may for brevity write, • ^=£'^=1 (^)-,: Now reasoning as in Chap. Vlll., the effect of the elimination will be the same if a and h, instead of being constants, are 22—2 • 840 NON-LINEAR EQUATIONS OF THE FIRST [CH. XIV, made functions of x and y, so determined as to preserve to the equations (9) their actual form. But a and h being made variable, we have •^ dx da dx db dx * _ df df da dfdh " dy da dy db dy ' Hence the equations for determining a and h are ^da^dfdb^^ da dx db dx ' df da df dh _ , . da dy db dy ^ Now this system may be satisfied in two distinct ways, 1st by assuming !=«' %-' (^2)- The values of a and h hence found lead, on substitution in the complete primitive, to that solution which Lagrange terms singular. df „^j df 2ndly, Supposing — and -^ not to vanish, we have, on (Xa (to elimination of them from (10), (11), da db da db _^ ,^ «v dx dy dy dx Now this supposes either, 1st, that a and b are constant, which leads us back to the complete primitive ; or, 2ndly, that b is an arbitrary function of a. Chap. II. Art. 1. Again, multi- plying (10) by dx and (11) by dy, and adding, we have ^da + ^db = (14). da db ART. 8.] ORDER WITH THREE VARIABLES. 841 Thus tlie system (10), (11) is now replaced by the system (13), (14). Making then, in accordance with (13), 6 = ^ (a), the expres- sion for z in (8) becomes while (14) becomes And these together constitute what Lagrange terms the gene- ral primitive. To apply them it is only necessary to give a particular form to ^ (a), and then eliminate a. Hence the following theorem. Theorem. A complete j^nmitive of a partial differential equation of the first order being expressed in the form ^=f{^i y, «; ^) (15), the general primitive will he obtained hy eliminating a between the equations ^^ df{x,y,a,^(a) } > (16), da J the singular solution, by eliminating a and h between (15) and the equations df(x, y, a,b) ^^ df{x,y,a, h) ^^ ,^^. da ' db It will be observed that the process for obtaining the general primitive is virtually equivalent to that by Avhich we should seek the envelope of the surfaces defined by the corresponding complete primitive, the constants a and b being treated as variable parameters connected by an arbitrary relation, while the process for obtaining the singular solution is that by which we should seek the envelope of (15), supposing a and b to be independent parameters. 34^ NON-LINEAR EQUATIONS OF THE FIRST [CH. XIV. Thus, of the system of solutions which consists of a complete primitive, a general primitive, and a singular solution, the complete primitive must be regarded as forming the basis, and the system itself geometrically interpreted includes the surfaces represented by the complete primitive together with the whole of their possible envelopes. Ex. To deduce the general primitive and singular solution of the equation z = pq, A complete primitive being z = (y + a)(x + b) (a), the corresponding general primitive will be expressed by the system 2={y + a){x + (l)(a)} = x + (a) is assigned. Another form of the complete primitive being (c^ + ^ + ef ^ = i W' the corresponding form of the general primitive will be Z=l{cx-\-^ + ylr (c)}' c = tc-^^ + f'{c) ^ W, from which c must be eliminated when the form of ^jr (c) is assigned. To deduce the singular solution, we have from (a), ~Y- = x -\-h = 0, da ART. 9.] ORDER WITH THREE VARIABLES. 343 Hence, h= — x, a=^ — y which, substituted in (a), gives s =i 0, a singular solution. The same result is deducible from (c). 9. In the last example, two complete primitives, two cor- responding forms of general primitive, and one common form of singular solution are presented. Two systems of solution appear, and the question arises : Does either system suffice alone ? The answer is given in the following theorem. Theorem. All possible solutions of a partial differential equation of the first order, are virtually contained in the system consisting of a single complete primitive j with the derived gene- ral primitive and singular solution. As before, we shall represent the proposed differential equation and its given complete primitive in the forms, F{x,y,z,p,q) = (18), ^=f{^, y, «, &) (19)- We shall also represent the form, ^ = X(^,2/) (20), some solution of (18), of which nothing more is known than that it is a solution. We are to shew that such solution is included m the system of solutions of which the common primitive (19) constitutes the basis. If we represent for brevity the values of z in (19) and (20) by y and % respectively, we shall have, since both are solu- tions of (18), ^(-'^'/'l'|)=« ^21), From the form of the above equations it appears that if a and 6 are so determined as to satisfy two of the conditions, J"^' dx~~dx' dy dy ^ ^' 844 NON-LINEAR EQUATIONS OF THE FIRST [CH. XIV. they will satisfy the third. For suppose they satisfy the first two, then the system (21), (22) maybe expressed in the form in which the truth of the third equation of (23) is involved. Now, as (19) satisfies (18) whatever constant values we assign to a and h, it still will do so if, after the differentiations by which -,— and -y- are found, we substitute for a and h •^ ax ay any functions of x and y. But a and h can be determined so as to satisfy two con- ditions. Hence they can be determined so as to satisfy the system (23). Differentiating the equation /= ;)^ on the h3'po- thesis that a and h are functions so determined, we have df df da df db _ dx dx da dx db dx dx ' df df da dfdb _dx dy dady db dy dy ' df df Here, -;-, -— have the same values as in (23), being ob- tained by differentiating as if a and b were constant. Hence, reducing by (23), we have dfda dfdb _ ^1 da dx dbdx I dfda dfdb _ da dy db dy (25). But these are the equations (10), (11), Art. 8, by which the system of solutions founded upon the complete primitive is constructed. The argument then is briefly this. If ^ = ^ (x, y) is a solution of the given partial differential equation, it is possible to determine a and b in the given comj)lete primitive so as to satisfy the equations ^23) ; therefore so as to satisfy the ART. 9.] ORDER WITH THREE VARIABLES. 345 equations (25) ; therefore so as to indicate a necessary in- clusion of 2 = x (^' I/) i^ ^^^ system which is founded upon the given complete primitive. Cor. 1. Hence the connexion of a given solution with a given complete primitive may be determined in the following manner. Adopting the foregoing notation, determine the values of a and h which satisfy the system (23). If those values are constant, the solution is a particular case of the complete primitive ; if they are variable, but so that the one is a function of the other, the solution is a particular case of the general primitive ; if they are variable and unconnected it is a singular solution. Cor. 2. Hence also any two systems of solutions founded upon distinct complete primitives are equivalent. For each is virtually composed of all possible particular solutions. Ex. The equation js =pq, has for its complete primitive z = (x + a) {y + h), and for a particular solution z= . - What is the connexion of this solution with the complete primitive ? We have by (23), 7 y + x y + X These equations are not independent, the first being the product of the last two. Any two of them give ^ y — X T x — y 2 * 2 ' whence 'b — — a. Thus, the values of a and h being variable, but such that 5 is a function of a, the proposed solution is a particular case of the general primitive. Some general questions, but of minor importance, relating to the functional connexion of different forms of solution, will be noticed in the Exercises at the end of this Chapter. 346 DERIVATION OF THE SINGULAR SOLUTION [CH. XIV. In quitting this part of tlie subject, we may observe that there are two modes in which the questions it involves may be considered. The first consists in shewing- that the o-ain of generality, which in Charpit's process accrues in the trans- ition from the complete to the general primitive, is equal to that which Lagrange's original but far more difficult process secures by the employment of the general value of j) drawn from (4), instead of a particular value drawn from its auxiliary system. The proof of this equivalence, as developed with more or less of completeness, by Lagrange and Poisson {Lacroix, Tom. IL p. 564, iii. p. 705), and recently by Prof. De Morgan {Camhridge Journal, Vol. Vll. p. 28), is, from its complexity, unsuitable to an elementary work. The other mode is that developed in the foregoing sections. Derivation of the singular solution from the differential equation^ 10. The complete primitive expresses z in terms of x, y, a, h. The differential equation expresses z in terms of x, y^ p, q. Either is convertible into the other by means of the two equations derived from the complete primitive by differ- entiating with respect to x and ?/ respectively. Hence it is not difficult to establish the two following equations, dz d^z dz d^z dz da dhdy db dady dp d^z d^z d^z d^z dadx dhdy dady dhdx dz d^z dz d?z dz _ da dhdx db dadx dq^ ITz d^z dz S^ (26), dadx dhdy dady dhdx in the first members of which z is supposed to be expressed in terms of x, y, p, q by means of the differential equation, in the second members, in terms of x, y, a, h by means of the complete primitive. ART. 11.] FROM THE DIFFERENTIAL EQUATION". 347 Now the singular solution is deduced from the complete primitive by means of the equations l^«' S=o (2")= and it is evident from the form of (26), that this will gene- rally involve the conditions ^ !=«' !=<>••• • (2«)- Such then will generally be the conditions for determining -the singular solution from the differential equation. The conditions (28) will not present themselves, should the denominator of the right-hand members of (26) vanish identi- cally. But it may be shewn that in this case the conditions (27) do not lead to a singular solution. And analogy renders it probable that whenever the conditions (28) are satisfied the result, if it be a solution at all, will be a singular solution. The complete investigation of this point, however, would in- volve inquiries similar to those of Chapter viil. The Rule indicated is then to eliminate p and qfrom the differential equation by means of the equations (28) thence de- rived. [See the Supplementary Volume, Chapter xxiv. Art. 8.] 11. The following geometrical applications are intended to illustrate the preceding sections. Ex. 1. Required to determine the general equation of the family of surfaces in which the length of that portion of the normal which is intercepted between the surface and the plane x, y, is constant and equal to unity. As the length of the intercept above described in any sur- face is z (1 ■\-'p^ + 2^)2, we have to solve the equation ^'(1+/ + 2') = 1 W- Hence q = (z~^ — 1 —p^y^, and the auxiliary system (5), Art. 7, becomes, on substitution and division by {z~^ — 1 —py-^, dx _ dy _ ^^ _ ^^V fL\ p"' (r^-1-/)^ ~ 5-^- 1 ~"" "y • ^^' 848 DEEIVATION OF THE SINGULAR SOLUTION [CH. XIV- From the last two members we have on integration c(l-/)^ z Substituting this, with the corresponding value of g derived from (a), in the equation dz =jpdx 4- q^dy we have z ^ ^ z integrating which in the usual way, we find or, changing the signs of c and c', {\-z'f^cx-{l--efy-rd (c), which is a complete primitive. The corresponding form of the general primitive will be o=^+c(i-cT^?/+^'(c)J ^ ^' from which c must be eliminated. But another system of solutions exists ; for from the first, third, and fourth members of (5) we may deduce 'pdz + zd^ 4- c?^ = 0, whence 'pz •\- x — a, from which, and from the given equation determining j; and ^, we have to integrate , a — x^ \\ — [a — ocf — z^\'^ ^ dz=^ dx + ^ ^ ^-dy. Z Z '^ The result is {x-ar+{y-iy + z'=l .(e), a complete primitive. The corresponding general primitive is x-a+[y-i-{a)]^'[a)=Qy] ; ^■' '' ART. 11.] FROM THE DIFFERENTIAL EQUATION". 349 To deduce the singular solution from the differential equa- tion (a) we have whence p = 0, q=0; substituting which in (a) we find z = ±l. The above example illustrates the importance of obtaining, if possible, a choice of forms of the complete primitives. The second, of those above obtained, leads to the more interpret- able results. It represents a sphere whose radius is unity and whose centre is in the plane oo, y, while the derived general primitive represents the tubular surface generated by that sphere moving but not ceasing to obey the same conditions. The singular solution represents the two planes between which the motion would be confined. All these surfaces evidently satisfy the conditions of the problem. Ex. 2. Required to determine a system of surfaces such that the area of any portion shall be in a constant ratio {m : 1) to the area of its projection on the plane cot/. The differential equation is evidently 1 + p^ 4- q^ = m^, and it will readily be found that it has only one complete primitive, viz. 2 = ax + fj{m^ ~ a^ — 1) j^ -j. ^, Thus the general primitive is z = ax-\- Aj{m^ — a^ — 1) y + ^ (a), and this represents various systems of cones and other develop- able surfaces. 350 SYMMETRICAL AND GENERAL SOLUTION [CH. XIV. Similar but more interesting applications may be drawn from the problem of the determination of equally attracting surfaces. 12. Attention has already been directed to the different forms in which the solution of a non-linear equation may sometimes be presented. It may be added that linear equa- tions admit generally of a duplex form of solution. The ordi- nary method gives directly the equation of the system of surfaces which they represent ; Charpit's method leads to a form of solution which exhibits rather the mode of their genesis. Ex. Lagrange's method presents the solution of the equa- tion {mz — ny)]p-{-{nx—lz)q = ly — mx (a), in the form Ix -{■ my ■]- nz = <^ {x^ -\- y^ -^ z^) (h), the known equation of surfaces of revolution whose axes pass through the origin of co-ordinates. ■ Charpit's method presents as the complete primitive of (a) {x-clf ■\- {y — cmy+ [z — crif — r^ (c), c and r being arbitrary constants. This is the equation of the generating sphere. The general primitive represents its system of possible envelopes. These solutions are manifestly equivalent. Bymmetrical and more general solution of partial differential equations of the first order. 13. The method of Charpit labours under two defects. 1st, It supposes that from the given equation q can be ex- pressed as a function of x, y,z,p', 2ndly, It throws little light of analogy on the solution of equations involving more than two independent variables— a subject of fundamental import- ance in connexion with the highest class of researches on Theoretical Dynamics. We propose to supply these defects. . APvT. 13.] OF PAKTIAL DIFFEEENTIAL EQUATIONS. S51 It will have been noted that Charpit's method consists in determining _p and ^ as functions of x, y, z, which render the equation dz =pdx + qdy integrable. This determination pre- supposes the existence of two algebraic equations between X, y, z, p, q; viz. 1st, the equation given, 2ndly, an equation obtained by integration and involving an arbitrary constant. Let us represent these equations by F{x,y, z,p, q) = 0, <^ {x, y, z, p, q) = a.,.(29), respectively. And let us now endeavour to obtain in a general manner the relation between the functions i^ and . Simply differentiating with respect to x, y, z, p, q, and re- dF , ^ d^ , ,., dF , _. d(p , ^, , presentmg -^-^ by X, ^ by A , ^ by P, ^byP,&c. we have Xdx + Ydy + Zdz + Pdp -\- Qdq = 0, Tdx + Y'dy + Z'dz + Fdp + Q'dq = ; or, substituting^Ja? + qdy for dz, {X-{-pZ)dx+{Y+qZ)dy + Pdp + Qdq=0....(SO), {X-^pZ') dx + (r + qZ') dy + Fdp + qdq = 0....(31). d z dj z d"z But, representing for brevity ^, , ^^ and — :, , by r, s, t, respectively, we have dp — rdx 4 sdy ] dq = sdx + tdy ) ^ (32). Substituting these values in (31) we have (X'-^pZ'+ tF' + sQ) dx-Y{T + qZ-\-sF+tQ') dy=0, which, since dx and dy are independent, can only be satisfied by separately equating to their coefficients. These furnish then the two equations -{X'-vpZ')=TF + sQ-\ ,33. . 1 _(r+2^') = s-P'+w 852 SYMMETRICAL AND GENERAL SOLUTION [CH. XIV. Now these equations are of the same fovm as (32). They establish the same relations between the functions -(X!^pZ% -{Y' + qZ'), F, Q', (34), as (32) does between the differentials dp, dq, dx, dy. It follows that if we give to dx and dy, which are arbitrar}'-, the ratio of the last two of the functions (34) then will o?p and dq have the ratio of the first two, so that the following will be a consistent scheme of relations, viz. dx _dy _ ^P _ ^^ /o -N F~4^~"X+p^'~~" T + qZ' ^ '^* Now dividing the successive terms of (30) by the successive members of (35) we have [X-\-pZ) P' + ( F+ qZ) Q'-P {X' +pZ') -Q{r+qZ'):=0 (36). This is the relation sought. It might be obtained by direct elimination by multiplying the equations of (33) by P and Q respectively, and the corresponding equations derived from (30) by P and Q' respectively, and subtracting the sum of the former from the sum of the latter. It is obvious too, and the remark is important, that we might pass directly from (30) to (36) by substituting for dx, dy, dp, dq, the functions of (34), and that this substitution is justified by the identity of relations established in (32) and (33). If in (36) we substitute for X, Y, &c. their values, and transpose the second and third terms, we have /dF dF\d^_fdj^ d^\dF fdF dF\d^ \dx ^ dz 1 dp \dx -^ dz J dp \dy ^ dz J dq fd^ d^\dF ^ ,_^, -(^ + 5^7)5^=" (•^7)- Such is the relation which connects the functions F and . When F is given it assumes the form of a linear partial differ- ART. 14.] OF PARTIAL DIFFERENTIAL EQUATIONS. S53 ential equation of the first order for determining ^. If from its auxiliary system we can deduce any integral involving an arbitrary constant, and such, that in conjunction with the given equation it enables us to determine p and q as functions of X, y, z, the subsequent integration of dz = ]pdx -\- qdy will lead to a form of the complete primitive. 14. Analogy now points out the method to be pursued for the solution of equations involving more than two inde- pendent variables. Prop. To deduce the complete primitive of the partial differential equation F {x^, x^, ,.. x^, z, p^, p.^, ,.,p^ ^0 (38), , dz dz where p= ..p^^ 1 n In the first place we must seek to determine values of Pj, p^y-'-Pn i^ terms of the primitive variables x^, x^,... x^, Zj such as will render integrable the equation dz = p^dx^-\-p^dx^ '"+Pndx^^ (89). Suppose one of the equations requisite in conjunction with (38) for this determination to be ^ {x^,x^, ... x^, ^,p„p„ ...pj =a^ (40). Then representing the first members of (38) and (40) by their characteristics jPand , differentiating, and substituting for dz its value given in (39), we have results which may be thus expressed, {(£+^'S)^'^'+-rf|'^^'}=^ ^^^' where S^ represents summation from i—1 to ^ = n. dz But since ^^ = y- , we have B. D. E. 23 2, S54 SYMMETRICAL AND GENERAL SOLUTION [CH. XIV. Substituting this value in (42), we shall be permitted, in con- sequence of the independence of the differentials dx^, dx^,. . .doc^, to equate their respective coefiQcients to 0. It is easy to see that the coefficient of dx^, will be d^ d^ ^ d^ d^z dx^ *" dz ' dpi dxidxy ' Equating this to 0, we have, on transposition, fd^ d^\ _y d^z d^ \dxr dz J ' dXidxj, dp.^ ' Hence, changing { into r and r into i, ._f^* + «.^U2 _^f_d^ (44) \dxi -^^^ dzj *" dxj.dXi dpj. Now comparing this with (43), and observing .that d^z _ d'^z we see that the sy^ems of differentials represented by dpi and dx^ respectively are connected by the same relations as the systems of functions represented by ^d<^ d^\ , d^ d^\ d^ -\-Pi-T-j and -y- respectively. \dxi ^' dz J dp^ Hence, by the reasoning of the previous example, it is per- mitted to substitute in (41)^for the differentials, the correspond- [d^ d^\ d*^ ing functions, viz. — \'j~'^Pi~T~) for dp)i', and -y- for dx^. \Ye thus find {(dF dF\d^ dF/d'S? ^*\1 _ „ Kcte, ^^'' dz j dp, dp, \dx, +^' di)]~^ ^*^^' ART. 14.] OF PARTIAL DIFFERENTIAL EQUATIONS. 355 the summation extending from { = 1 to i== n. This is the relation sought, and it is seen to be symmetrical with respect to F and O. When F is given, it becomes a linear partial differential equation for determining . From its auxiliary system of ordinary differential equations it suffices to obtain 71 — 1 integrals, such as, in conjunction with the given equation, will enable us to determine p^,p.^, ... ^„ in terms of the original variables; then integrating (39), we shall obtain the complete primitive in the form f{x^,x^,,,.x^y z, a^,a^, ... aj = (47). All other forms of solution are hence deducible by regarding a,, ^2, ... a^ as parameters varying, independently or in sub- jection to connecting relations, but so as to leave unaffected the forms of p^, p^, ... Pn* It is proper to observe that the given equation F= is itself included among the particular integrals of (45). In fact F is one of the forms of ^ which make = a a solution, as will be found on trial. The given equation is therefore a particular integral. And therefore the n—1 integrals of the system (46) must be independent of it in order to render the determination of^^, p^, "-Pn possible. The equation (45) may be expressed as follows: ^ fdFd^_dFd^\ dF^ ^_^^ ^-0 * \dXi dpi dpi dxj dz '^' dpi dz '-^^ dpi And under this elegant form, obtained however by a more complex analysis, the solution is presented by Brioschi {Tor- tolini, Tom. vi. p. 426, Intorno ad una proprietd delle eqiia- zioni alle derivate parziali del primo or dine). The problem of the integration of partial differential equa- tions of the first order, irrespectively of the number of the variables, appears to have been first solved by Pfaff, but the 23—2 356 SYMMETRICAL AND GENERAL SOLUTION [CH. XIV. most complete discussion of it will be found in a memoir by Caucliy {Exercices d' Analyse, Tom. ii. p. 238. 8ur Vintegra- tion des equations aux derivees partielles du premier ordre), in which the determination of the arbitrary functions of the general primitive so as to satisfy given initial conditions is fully considered. The connexion of the subject with Theo- retical Dynamics was first established by the researches of Sir W. Hamilton and Jacobi. The truth, illustrated above, that the solution of a partial differential equation of the first order is reducible to that of a system of ordinary differential equations, and the truth that the solutions of certain systems of differential equations (including that of dynamics) may be reduced to the discovery of a single function defined by a partial differential equation, are correlative. The researches above referred to, together wdth those of Liouville, Bert rand, and Bour, founded partly upon their results and partly upon the allied discoveries of Lagrange and Poisson concerning the variation of the arbitrary constants in dynamical problems, contain the most important of recent additions to our specu- lative knowledge of Differential Equations. For this reason we have dwelt upon their history. Fuller information will be found in Mr Cayley's excellent Report on the recent Pro- gress of Theoretical Dynamics. {Report of British Associa- tion, 1857.) [In an Appendix to the first edition Professor Boole pre- sented Art. 14 in the following form.] Art. 14. The most important form of the problem of this Article is the following, and the reader is requested to sub- stitute it for the one in the text, sufficient account not being there taken of the conditions among the constants. Eequired a value of ^ as a function of x^, x,^, ... x^ which shall satisfy the partial differential equation F{x^,x^,...x^, z,p^,p^,...pj = (1), and shall, when x^ = 0, assume a given form, z = "-P»!-i' ^^ may express, and so replace, these arbitrary constants by initial values of the above vari- ables corresponding to x^ = 0. Let ^1 , . . . f„_i, f, vr^, . . . TT^j.i be the new constants in question ; then, substituting these for the variables whose initial values they represent in the n equations (2), (3), we obtain n con- ditions connecting the above constants. Thus we have finally 37^ equations, consisting of 2n inte- grals with n equations of condition connecting the 272 — 1 constants which those integrals contain. From these 3?2 equations we can eliminate the above 2n — 1 constants toge- ther with the n quantities p^,p^, .,. p^. The result will be a final relation between^, x^, x^, ••• ^„) which will be the solu- tion sought. If we regard the functioned {x^, x^, ... ic„_J as arbitrary, the above solution will constitute a general primitive ; but if we give to it a particular form involving n arbitrary constants, we shall obtain a complete primitive. (Cauchy, Exercices, Vol. II. p. 238.) [Important additions on partial differential equations of the first order are given in the Supplementary Volume^ Chap- ters XXV., XXVI., and xxvii.] 358 EXERCISES. [CH. XIV. EXERCISES. 1. How are equations, in wliicli all the differential coefB- cients have reference to only one of the variables, solved ? 2. ^^^ y , o ^^ _ y dx x-\- z 4. The partial differential equation of the first order which results from a primitive of the form u=f{v), where u and v are determinate functions of x, y^ z, is necessarily linear. Prove this. 5. ap-\-lq^ = l, ■^ ^ a 7. yp + xq — z, 8. x'^p — xyq +2/^ = 0. 9. Integrate the equation of conical surfaces (a — x)p + (h—y)q = c—z. 10. xzp + yzq = x7/, 11. (^' + 2' - x'')p - 2xyq + 2xz = 0. 12. Required the equation of the surface which cuts at right angles all the spheres which pass through the origin of co-ordinates and have their centres in the axis of a?. It will be found that this leads to the partial differential equation of the last problem, 13. z-xj>-'yq = a{x^+y^-\-z'')K CH, XIV.] EXERCISES. S59 14. Find the equation of the surface which cuts at right angles the system of ellipsoids represented by the equation where D is the variable parameter. Lacroix, Tom. ii. p. 678. 15. Find the equation of a surface which belongs at once to surfaces of revelation defined by the equation pi/ — qx = 0, and to conical surfaces defined by the equation po) -\-qy = z. In problems like tlie above we must regard the equations as simultaneous, determine p and q as functions of x, y, z, and substitute their values in the equation dz =pdx + qdy, which will become integrable by a single equation if the problem is a possible one, but not otherwise. ^^ dz dz ,d2 xy Id. a; -^ -}- V -y- + ^ ^7 = ^-^ + ~f • ax *^ ay dt^ t 17. Explain the distinction between a complete primitive and a general primitive of a partial differential equation of the first order. 18. Find the complete and the general primitive of z = px + qy -\- pq. 19. Deduce a singular solution of the above. 20. pq=l, 21. q = xp-{-p\ 22. Shew from the form of its integral that q[=f(p) belongs only to developable surfaces. 23. Deduce two complete primitives of ' pq=px + qy. 24. Deduce two complete primitives of «> 60 EXERCISES. [CH. XIV. 25. Given two general primitives of a partial differential equation of the first order, in the forms, 1st. . = F[.,y,a,4>ia)], o = ^^} , 2nd. . = *h2/,c,^(c)}. o = '^^^^^l^±^K shew that the dependence of the functions ^}r (c) and (a), when the two primitives lead to the same particular integral, may be determined by the following rule. Eliminate x and y from any four independent equations of the system F_fl) ^_^ dF_d^ ^-0 — -0 ^ dx dx' dy dy ' da ■ do The two resulting equations will involve the relation required, and when the form of (a) is given, the elimination of a from both will give a differential equation for determining the form of i|r (c). 26. The equation z—jpq^ has two general primitives, 1st. z:={y + a)[x + (i>{a)l =^ -^[{y ^ a] [x + j> [a)]], 2nd. 42 = {c^+^ + t(c)r, 0=;^[c^ + ^ + ^(c)f; c dc shew hence that the relation between ^ {a) and ^^ (c) is ex- pressed by the equations f(a)+i=0, c^[c)-c'f'{c) = 2a. c [An interesting problem involving partial differential equa- tions of the first order is discussed in the Supplementary Volume, Chapter xxxiii.] ( 861 ) CHAPTER XV. PAETIAL DIFFEEENTIAL EQUATIONS OF THE SECOND ORDEE. 1. The general form of a partial differential equation of the second order is where F(x, y, z,p, q, r, s, t) = (1), dz dz d^z d"z d^z ^ dx^ dy^ do^ * dxdy ' dif ' It is only in particular cases that the equation admits of integration, and the most important is that in which the dif- ferential coefficients of the second order present themselves only in the first degree; the equation thus assuming the form Rr + Ss+Tt=^V (2), in which B, S, T and V are functions of x, ?/, z, p and q. This equation we propose to consider. The most usual method of solution, due to Monge, consists in a certain procedure for discovering either one or two first integrals of the form «=/W • (3), u and V being determinate functions of x, y, z, p, q, and / an arbitrary functional symbol. From these first integrals, singly or in combination, the second integral involving two arbitrary functions is obtained by a subsequent integration. An important remark must here be made. Monge's method involves the assumption that the equation (2) admits of a first integral of the form (3). Now this is not always the case. There exist primitive equations, involving two arbitrary func- tions, from which by proceeding to a second differentiation both functions may be eliminated and an equation of the form (2) obtained, but from which it is impossible to eliminate 362 PARTIAL DIFFERENTIAL EQUATIONS [CH. XV. one function only so as to lead to an intermediate equation of the form (3). Especially this happens if the primitive in- volve an arbitrary function and its derived function together. Thus the primitive ^ = ^(y + ^) + i/r(y~a?)-^{'(?/ + ^)-'^'(3/-^)}...W, leads to the partial differential equation of the second order r-« = ^ (5), X but not through an intermediate equation of the form (3). It is necessary therefore not only to explain Monge's method, but also to give some account of methods to be adopted when it fails. [Part of the present Chapter is treated on a larger scale in the Supplementary Volume, Chapters XXVIII. and XXIX.] 2. It is not o»nly not true that the equation (2) has neces- sarily a first integral of the form (3), but neither is the con- verse proposition true. We propose therefore, 1st, to inquire under what conditions an equation of the first order of the form (3) does lead to an equation of the second order of the form (2) ; 2ndly, to establish upon the results of this direct inquiry the inverse method of solution. And this procedure, though somewhat longer than that usually followed, is more simple, because exact and thorough. Prop. 1. A partial differential equation of the first order of the form u—f[v) can only lead to a partial differential equation of the second order of the form Br+Ss+Tt=V (6), tvhen u and v are so related as to satisfy identically the con- dition du dv du dv _ ^ ,^ dp dq dqdp ^ ^' For, differentiating the equation u =f{v) with respect to x, and observing that -r^ = j), -^ = r, -j^ =s, we have ART. 8.] OF TSE SECOND ORDER. 363 du du dii du_j.,,.fdv dv do dv\ dx -^ dz dp dq "^ \dx ^ dz dp dqj * In like manner differentiating u =f{v) with respect to y, we have du du du du _ .,. . fdv dv dv dv\ dy ^ dz dp dq -^ ^ ^ \dy ^ dz dp dqj ' EKminatingy(v) there results du die du du\ fdv dv dv dv dx ^ dz dp dq) \dy ^ dz dp dq fdv dv dv dv\(du> du du du\ ^ \dx ^ dz dp dq)\dy ^ dz dp dq)~ '"^ '' On reduction it will be found that the only terms involving r, s, and ^ in a degree higher than the first will be those which contain rt and s^. The equation will in fact assume the form Rr ^ Bb^- Tt'\- U{rt - ^^) = V. (9), • 1*1 TT ClU diV ecu CCV rni p p 1 1 1 1 m which ty=-i — -, r- -r~ * -Lhe lorms oi the other co- ap dq dq dp efficients it is unnecessary to examine. Now this equation assumes the form (6) when the con- dition (7) is satisfied — and then only. 3. The proposition might also be proved in the following manner. Since io=f(v) we have du=f'(v)dv, an equation which, since f(v) is arbitrary, involves the tT\^o equations dit = 0, do= 0. Hence du , d?i ^ du ^ du ^ du , -i-dx+ -f- dy + -r- dz + -y- dp + -j- dq = ax dy "^ dz dp *■ dq ^ dv , dv , dv ^ dv ^ dv ^ -Y- dx -\- -r du -\- -^ dz ■\- -J- dr) ■\- ^r dq — \) dx dy "^ dz dp ^ dq ^ \...m- S64 PAKTIAL DIFFERENTIAL EQUATIONS [CH. XV. But dz = pdx + qdif, dp = rdx + sdy, dq = sdoD + tdij. Whence on substitution fdii du du du\ , (du du ^ du ^ du\ , ^ fdv I dv do dv\ , fdv dv ^^l/^"^^ _0 \dx ' ^ dz dp dq) \dy ^ dz dp ' dq) Whence eliminating dx and dy, we have the same result as before., 4. A consequence, which, though not affecting the present inquiry, is important, may here be noted. It is that it would be in vain to seek a first integral of the form it =f{v) for any partial differential equation of the second order which is not of the form (9j. Prop. 2. To deduce when possible a first integral, of the form u =f(v), for the partial differential equation (6). By the last proposition u and v must satisfy the condition (7), which is expressible in the form, da du dv dv . dq 'dp dq ' dp ^ ^* Hence, if we represent each member of this equation by m, we have du _ du dv dv . ». dq dp' dq dp Substituting these values in (10), we have du 7 du , du ^ du , -, ^ , -J- dx -\- -J- ay + -J- dz -\- -T- [dp + mdq) = ^...(13); dv ^ dv ^ dv y dv , -, , . ^ , -,— aw + -J- du + -y- dz + -Y- (dp + m-dq) = ax dy ^ dz dp ^ -^ -^^ J and we are to remember that this system, being equivalent to du = 0, dv = modified by the condition (7), can only have an- integral system of the form, u^a, V = 6 ..(14), ART. 4.] OF THE SECOND ORDER. 865 a and h being arbitrary constants, and u and v connected by the condition (11). Making dz =pdoc -\- qdy in (13), we have 'du du\ , /du du\ -, du fdu du\ , fdu du\ ^ du , ^ y s r. /dv dv\ , fdv dv\ , dv , ^ 7 \ /^ ...(15). From these and from the equations dp — rdx + sdy, dq = sdx + tdy (16)> if we eliminate the differentials dx, dy, dp, dq, we shall necessarily obtain a result of the form (6). For in thus doing we only repeat the process of Art. 3, with the added condition (7). To effect this elimination, we have from (16), dp + mdq = (r + ms) dx-\- {s + mt) dy ; or, rdx + s [dy + 'mdx) + tmdy = d[p -f mdq (17) • Now the system (15) enables us to determine the ratios of dy and dip + mdq to dx, and these ratios substituted in (17), reduce it to the form (6). But in order that it may be, not only of the form (6), but actually equivalent to (6), it is necessary and sufficient that we have - dx _ dy 4- TYidx _ mdy _djV +rndq . . 'E^^~E ~t'^ F ^ ^' This system of relations among the differentials must thus include the equations (15). The same system (18), together with the equation dz =pdx + qdy, must therefore include the system (13). It must therefore in its final integral system include the equations u = a, v = h with their implied con- dition. 366 PARTIAL DIFFERENTIAL EQUATIONS [CH. XV. We conclude then, that if the equation Rr -{■ Ss+ Tt= F, result from an equation of the first order of the form u =f {v), the system (18), together with the equation, dz =pdx + qdy (19), must admit of an integral system determining u and v in equations of the form u = a, v = b. To eliminate m from (18) we have, on determining its value from the first and third members, substituting it in the second and fourth, and reducing, Mf-Sdxdi/ + Tdx'=0 (20), Rdpdy + Tdqdx — Vdxdy = (21), and these, with (19), make three ordinary differential equations among the five variables x, y, z, p, q. But among five vari- ables there ought to exist four ordinary differential equations in order to render the final relations determinate. And this confirms what was said in Art. 1, of the hypothetical character of Monge's method. It is only when the proposed equation originates in an equation of the form u=f(y), that the above system admits of two integrals of the form, u = a, V = h. As (20) is of the second degree it will, unless it is a com- plete square, be resolvable into two equations of the first degree, and either of these in conjunction with (21) and (19) may lead to a final integral system determiniug u and v. It follows that when the given equation admits of a first integral at all, it will admit of two such — excepting the case in which (20) is a complete square. 5. As yet no account has been taken of the quantity m. The mode in which it is involved in the equation (18), leads however to a remarkable consequence developed in the follow- ing Proposition. Prop. If by the last proposition we obtain two first in- tegrals of the form u .=/W, «. = <^W (22), AET. 5.] OF THE SECOND OKDER. 367 and if, regarding these as simultaneous, we determine 'p and g as functions of x^ y, z, those values will be such as to render the equation dz = pdx -\- qdy integrable, and thus to lead to the second or final integral. For simplicity, we shall represent u^—f(v^ by F, and '^^2 ~ ^ W ^y ^' Thus the supposed first integrals are simply i^ = 0, ^ = (23). Now reverting to the system (18), and representing the ratio dy : dx by 7i, its first two equations assume the form, 1 ?i 4- m nm EST' and shew that m and n are the two roots of the equation Hence, the value of the ratio dy : dx corresponding to one of the first integrals (28), is the same as the value of m cor- responding to the other. Now for the value of m corresponding to the integral i^= 0, we have by definition, du^ di\ _ dg _ dq du^ dv^ dp dp du^ J,, . , d\\ __^g__dq du^ rr / s dv^ dp -^ ^ ^ dp dF __ dq ^7^1 " dp Again, seeking the value of the ratio dy \ doc^ correspond- ing to the integral <[> = 0, we have (24). 368 PARTIAL DIFFERENTIAL EQUATIONS [CH. XV. fd^ d^ d = 0, and of dy : dx corresponding to the in- tegral F=0, we have dFd^ dFd^ dFd^ dFd^ dx dp dy dq dz dp dz dq dFd^ (dFd^ dFd^\ dFd^ . dp dp \dq dp dp dq) dq dq Subtracting (25) from (26), there results dFd^ _dFd^ dFd^ dFd^ dx dp dp dx dy dq dq dy ,(dFd^_dFd^\ ,fdF_d^__dFdj^ ^^ ,^^. \dz dp dp dz ) ^ \dz dq dq dz ) ^ Now this is identical with the equation (87), Chap. xiv. Art. 13, expressing the very condition which must be fulfilled in order that the values of p and q given by i^=0, = 0, may render the equation dz = pdx + qd^y an exact differential. Hence the proposition is established. It is interesting to observe that the two first integrals stand in a certain conjugate relation. Each of them satisfies that partial differential equation of the first order and degree which ART. 6.] OF THE SECOND ORDER. 8G9 we should have to construct in attempting, by the process of Charpit, to integrate the other. Hence also, although the knowledge of both is desirable, that of either is sufficient to enable us to proceed by integration to the final solution. 6. The statement of Monge's method, as derived from the above investigation, is contained in the following Eule. Rule. The equation being Rr-\-Ss+Tt= V, form first, the equation Rdif - Sdxdy + Tdx^=0 (28), and resolve it, supposing the first member not a. complete square, into two equations of the form dy — m^dx = 0, dy — m^d'x = (29). From the first of these, and from the equation Rdfpdy+Tdqdx-Vdxdy = Q ..,...(30),. combined if needful with the equation dz =pdx + qdy, seek to obtain two integrals u^ = a, v^ = b. Proceeding in the same way with the second equation of (29), seek two other integrals u^ = a, v^ = /3, then the two first integrals of the proposed equation will be ^i=fiM> %=f2M (31). To deduce the second integral, we must either integrate one of these, or, determining from the two p and q in terms of X, y, and z, substitute those values in the equation dz = pdx + qdy, which will then satisfy the condition of integrability. Its solution will give the second integral sought. If the values of m^ and m^ are equal, only one first integral will be obtained, and the final solution must be sought by its integration. When it is not possible so to combine the auxiliary equa- tions as to obtain two auxiliary integrals u = a, v = h, no first integral of the proposed equation exists, and some other pro- cess of solution must be sought. B. D. e. 24 870 PARTIAL DIFFERENTIAL EQUATIONS [CH. XV. We may observe that the determination of ^ and ^ from the two first integrals is facilitated by the fact that u and v satisfy the condition (7). Interpreted by Chap. il. Art. 1, that con- dition implies thatp and q enter, in some single definite com- bination, in both u and v. Ex. 1. Given -y-r, - c^ ^— . = 0. ax ay Here R — 1, S=0, T = — a^, F= 0. Hence we have by (28) and (30), d'lf' — o?dx^ = 0, dpdy — a^dqdx = (a) . The former of these is resolvable into the two equations dy + ados =0, dy — adx = (5), of which the first gives y -{- ax = c, and at the same time reduces the second equation of (a) to the form dp + adq = 0, of which the integral is p-\-aq=G. Thus a first integral of the given equation is p -\- aq =

(y-^Cix) + 'ylr(y-ax) ^^ ^ ^[y + ax) -^^{y - ax) . Or (^ {y + ax) {dy + adx) — •yfr (?/ — ax) (dy — adx) dz = 2a Hence if ^|(/> (t) dt = ^, {t) and - ^Jf {t) dt = f^ (t), we have Here cp^, i^^ are arbitrary functions since cf) and i/r are such. ART. 6.] OF THE SECOND OEDEE. 371 It is seen that, in each of the first integrals, the condition (7) is satisfied, and assuming p+aq — (l)(7/ + ax) = F, p — aq — yjr {y — ace) = , it is easy to verify the condition (27). Ex. 2. Given r+ as + ht=^0. Proceeding as before, we find j> + 'nq = (j)(y— mx), p + mq = '^{y — nx), as the two first integrals of the proposed, m and n being the roots of the equation f — at + h=0. Hence, determining j? and q, substituting in the equation dz = pdos+ qdy, integrating and reducing we have z = <^^{y — mx) + 'v/^j (2/ — nx). But when m and n are equal we have only one first in- tegral, viz. p + mq — {y — mx). Treating this by Lagrange's process, we have the auxiliary system m 9 (^ — '^^) From the first two members we find y — mx — c. This enables us to reduce the equation of the first and third to the form 7 _ ^^ whence z = x(j){c) + c. Therefore, restoring to c its value, 2! — x (0) dx, whence 2/ + ^ (^) ^ = C'', or y + X([> {ax -\-hy + cz) — G\ Thus the final integral is 9/ + x(j) {ax -jrhy + cz) =yfr {ax + hy -\- cz). This solution may also be expressed in the form z = x(j)^ {ax -\-hy + cz) + y-^^ {ax -\-hy + cz), in which it is in fact presented by Monge, {Application de r Analyse d la Geometric, Liouville's edition, p. 79). The equation solved is that of surfaces formed by the motion of a straight line which is always parallel to a given plane, and always passes through two given curves. » 7. In the above examples V is equal to 0, and this always facilitates the application of Monge's method. The following is an example in which V is not equal to 0. Ex. 4. Given r — t= ^ x-\-y The auxiliary equations being 4w dy^ — dx^ = 0, d2:)dy — dqdx -j ^- dxdy = 0, one of the systems hence derived is 4r> dy—dx = 0, dp — dq-{ — dx = 0. There is also another system, but it is not integrable in the form u = a, v = h. From the first of the above equations we get y-x = a, ^i^-^5' + 2^I^=0. the latter of which may, since dz = pdx + qdx, be reduced to the form d {2y -a){p-q) + ^dz = 0, 874 PARTIAL DIFFERENTIAL EQUATIONS. [CH. XY. whence {2y — a)(j3 — g) -^ 2z = hy or, replacing ahj y — x, Hence a first integral of the proposed equation will be {x-¥y){p-q)+2z=f{y-x). ISTow this being linear, we have, by Lagrange's method, the auxiliary system dx __ — dy _ dz x-\- y x+ y f(y — a?) — 2z ' The equation of the first two members gives y + w = a, and this reduces the equation of the second and third to the form ' — di/ _ dz '^~f{2y-a)-2z' dz 2z_ f{2y-a) dy a a * or _2j whence z = — ^ | e "^ /.(% — a)dy + h. The final integral will therefore be found by substituting in the above, after integration, ^ + ^ for a, and F{y -\-x) for h. 8. Monge's method fails in so many cases, owing to the non-existence of a first integral of the assumed form it =f{v), that it becomes important to inquire how its defects may be supplied. And various methods, all of limited generality, have been discovered. Thus Laplace has developed a method applicable to all equations of the form Br + Ss+Tt + Pp+Qq + Zz = U; B, S, T, F, Q, Z, and U being functions of x and y only, — which consists in a series of transformations, each of which has the effect of reducing the equation to the form s + Fp+Qq + Zz=Uy ART. 9.] MISCELLANEOUS THEOREMS. 375 P, Q, Z and TJ being functions of x and ?/, to which each transformation gives new forms. It may be that among these successive forms, some one will be found which will admit of resolution into two linear equations of the first order. But there are probably no instances in which this method has been applied in which the solution may not be effected with far greater elegance, and with far greater sim- plicity, by the symbolical methods of the following Chapters. And even Laplace's method is better exhibited in a symbolical form. The subject will be resumed. See Chap. xvii. Art. 14. The following sections contain miscellaneous but important additions. Miscellaneous Themems^ 9. Poisson has shewn how to deduce a particular integral of any partial differential equation of the form P = {rt-syQ (45), where P is a function of j9, q,T, s, t, homogeneous with respect to the three last, n a positive index,, and Q any function of X, y, z, and the differential coefficients of z of any order,, which does not become infinite when rt — s^ = 0. Assuming g_ = 4* {p)i '^^^^ have s = 4,'{p)r, t = 4,'ip)s={{p), and (j>'(p), i.e. p, q, and -~- . Integrating this as an ordinary differential equation we obtain a relation between p, q and an arbitrary constant ; and this, integrated as a partial differential equation of the first order, gives the solution in question. Ex. Given r^-f = rt-s\ 376 ,miscella;n'eous theoeems. [ch. xt. Proceeding as above, we find 1 — {(j>' i^)Y = ; therefore 1 — {(j>'{p)Y = 0, whence -^ = +1^ ^= ±p + c; therefore ^ — oy = ^{y ± x), a particular integral. The above method is applicable to all equations of the second order which are homogeneous with respect to r, s, t, for then we have only to suppose Q = 0. 10. There exists in partial differential equations a remark- able duality, in virtue of which each equation stands con- nected with some other equation of the same order by relations of a perfectly reciprocal character. As respects equations of the first order the principle may be thus stated. Suppose that in the given equation rn_ 02 > a if = Wj\Z~^ — * ButX=p, ]r=g, therefore rf — 7 /I — ^^^ ~ ^^y ajj — race "p say — iDrn 02 j , , , 7 — Sdx + Rdy dq = sdx + tdy = j^rp_^-2 > whence, equating coefficients, _ T _ -8 ^ R AET. 13.] MISCELLANEOUS THEOREMS. 879 The extension of the theorem to higher orders involves no difficulty. 12. It is an immediate consequence of the above, that any equation of the form ^ [p, q) r-\-f{p, g)s-\-x{p,q)t = o (52) can be reduced to an equation of the form X{x,y)T-y\r{x, y) s -\- (f> {x, y) t = (53), usually more convenient for solution. Legendre's solution of the equation {l+q')r-2pqs+{l+p')t = 0, by the aid of the above transformation, will be found in Lacroix (Tom. ii. p. 623). The same transformation makes the solution of any equa- tion of the form Rr-\- Ss -\- Tt = V {rt - s') dependent on that of an equation of the form Rr+Ss+Tt= F, but with different coefficients. The subject of these trans- formations has been most fully treated by Prof. De Morgan (Cambridge Philosophical Transactions, Yol. viii. p. 606). 13. Legendre also shews how, by a transformation for- mally resembling the above, to integrate the equation r=f(s, t). Assuming s and t as independent variables, and v = sx + ty — q as dependent variable, the equation is reduced to the form dP + ^d^r^d?-" ^^*)' dr dr where S and T are the values of -y- and -j- furnished by the given equation. Lacroix, Tom. il. p. 631. 380 EXERCISES. [CH, XV. EXERCISES. 1. To what condition must u and v be subject, in order that u = f{v) may be a first integral of an equation of tiie iovm Br +88 + 1^=^1 Integrate by Monge's method the following equations : 2. ^V + 2x1/8 + yH = 0. 8. ^V — 2pgs +pH = 0. 4. Integrate ps — qo^^O. 5. Integrate by Monge's method the equation q {l + q)r— {p + q+ 2pq) s +^9 (1 +p) t = 0. 6. The solution of Ex. 3 may, by the law of reciprocity, be made to depend on that of Ex, 2. 7. Monge's method would not enable us to solve the 2» equation r — t= -^ , X 8. Deduce by Poisson's method a particular integral of (1 + ^') r - 2pgs + (1 +/) i5= a. 9. Shew that the equations H - s' =/(^, 5), and rt - s' = {/ [x, y)]~\ are connected by the law of reciprocity. 10. The solution of the equation r — t— [rt — s^) may be derived from that of the equation r — t -\ ^-- = 0. ^ X + y Art. 7, Ex. 4. . ( 381 )• CHAPTER XVI. SYMBOLICAL METHODS. 1. The term symbolical is, by a restriction of its wider meaning, applied more peculiarly to those methods in Ana- lysis in which operations, separated by a mental abstraction from the subjects upon which they are performed, are ex- pressed by symbols in whose laws the laws of the operations themselves are represented. Thus -T- is written sjrmbolically in the form -y- u, the sym- d . bol -7- denoting an operation of which %i is the subject. In thus expressing an operation by a symbol, in studying the laws of that symbol, and in founding processes and methods upon those laws, we introduce no strange or novel principle of Language; for it is the very office of Language to express by symbols the procedure of Thought. Thus also we may write du ( d \ ,-.s dhi, dii daf dx +^"=(i+"£ + ^)" (2)' and so on. It Avill be observed that the symbol precedes the subject on which it operates. Operations may be performed in succession. Thus (-J- + (2) \-r- -^rlAu \ax J \dx J •denotes that we 6rst perform on the, subject u the operation 382 SYMBOLICAL METHODS. [CH. XVI. d denoted by -r- + ^, and tlien on the result effect the operation d denoted by y- + a. Thus a and 6 being constant, we have f ^ + aV-^ + 6 U = f ;t- + a) (-^ + hu \dx J \dx J \dx J \dx d fdu ^ \ (dii . 7 dx \dx j \ax d u du When an operation is repeated, the number of times which it is understood to be performed is expressed by an index attached to the symbol of operation. Thus fd y fd \fd \ -7-+a u = (^i- ■\- a][-j~ + a)u \ax J \dx J \dx J — -^-^ + Z(2 -y- -{-au f-i). dx dx If in the second member of (3), as in the first, we separate the symbols from their subject, we have + '*)(i + *)"={i+^*+^)i+W"--^'^- Now the symbolic expressions for the equivalent operations ■ performed upon u in the two members of this equation are in formal analogy with the algebraic equation (m + a) (m + V) ii — {m^ + (a + h) m + ah] it, and this is a particular illustration of a general theorem to the statement and demonstration of which we shall now proceed. 2. If we compare the symbolical expressions jdx J \dx ^+a)(~-^ h^, ^-^(a + h)^-\-ab (6), AET. 2.] SYMBOLICAL METHODS. 883 whose equivalence is stated in (5), we see that each involves d -J- together with constant quantities. Each might therefore, to borrow the language of analogy, be described as a function d of -J- and constant quantities, or more briefly as a function of -J- , and expressed in the form / ( -7- ) • Again, each ex- presses a system of operations in the performance of which the presence of the symbol -j~ only indicates differentiation, not integration. We may with propriety term any function d . . . . d of y- possessing this character a direct function of -^ . The theorem in question is then the following. d Theorem. Any direct function of y- and constant quan- d titles may be transformed as if -7- were itself a quantity. In the first place it is evident that any direct function of d . the symbol -7- according to the above definition is, in form, d what we should term a rational and integral function of -7-, ax were that symbol merely algebraic. Now the laws, according to which algebraic symbols com- bine with each other in the composition of all rational and integral expressions, are the following, viz. 1st, the distributive, expressed by the equation m{u-\-v) = mu + mv (7), 2ndly, the commutative, expressed by the equation ma — am (8), Srd!}^, the index law, expressed by the equation mV = m"^' (93. 384 SYMBOLICAL METHODS. [CH. XVI. These determine, and alone determine, tlie forms, or, to speak more precisely, the permitted variety of form, of algebraic expressions of the above class. d But the symbol -y- , when employed in combination with constant quantities to operate on subjects which are not con- stant,^ is subject to laws formally agreeing with the above. For we have -j-au=a-j-u - (11), (i)°(iy"=(ip (''^' the last of these, however, expressing, not any distinctive pro- d perty of the operation -j- , but only the fact that it is an operation capable of repetition. These laws, in like manner, determine the possible forms of symbolic expressions involv- ed . ing -7- with constants, and representing direct systems of operations. Hence the variety of form permitted in the one case is the same as that permitted in the other. In other words, the same transformations are valid. Among the consequences of the above theorem the follow- ing may be noted. 1st, We can reduce any symbolical expression of the form d^^'^^' d^' "^ ^' d^' ... + «,, in which a„ a„ ,,.a^ are con- stants, to an equivalent expression of the form d _ \(d__ \ ( ^_ dx V \da> V '" \d.x ' ART. 3.] SYMBOLICAL METHODS. 385 where m^, m^, ••• '^« ^^^ ^^^ roots of the equation m" + «iW""' + «X"' • • • + ^,. = 0- 2ndly, The order, in which the component operations d d d dx ' dx ^' dx are written, is indifferent. d^u Ex. Thus --rj^ — a^u = may be reduced to either of the ctcc forms -r + ^][-7 a]u = 0, l-j a) ( -^ + a)^^ = 0. ax J \ax J \dx J \dx J 3rdly, The complex operation is itself, like the elementary operation [-j-] , distributive; i.e. representing that complex operation by / ( ^- ) , we have This conclusion may be verified, by substituting for /(-^ the expression for which it stands, and performing the operations. Inverse Forms, 3. All that is said above relates to the performance of operations, definite in character, upon subjects supposed to be given. But an inverse problem is suggested, in which it is required to determine, not what will be the result of perform- ing a certain operation upon a given subject, but upon what B.D.E. 25 386 INVERSE FORMS. [CH. XVT. subject a certain operation must be performed in order to lead to a given result. Thus, in the equation i + «)"=^ (1*)' d if u be given, the performance of the operation 'T- + cb deter- mines v\ but if V be given, then the inquiry arises, what is that unknown subject u, the performance of the operation -J- + a upon which will lead to the result v? As any procedure for determining ii from v is inverse to the procedure by which v is determined from w, analogy sug- gests the notation "=(i+") ^ (^•^^' -7- +«) representing the inverse procedure in question, but representing that procedure only in its inverse character, i.e. conveying no information as to how it is to be performed, but only telling us that it must be such, that if, having per- formed it on V, we perform on the result the operation -t- + ct, to which it is inverse, we shall reproduce v. For, substituting in (14) the expression for u given in (15), we have d \fd \-^ ax J \ax J The inverse procedure is thus presented as one, the effect of which the direct operation sirnply annuls. This is its definition. Thus in Arithmetic, division is inverse to multiplication. What is meant by dividing a by 6 is the seeking of a third number c, which when multiplied by h will produce a. And the very procedure by which this is effected consists not in any new and distinct operation for determining the subject c, but in a series of guesses, suggested by our prior general knowledge of the results of multiplication, and tested by multiplication. ART. 3.] INVERSE FORMS. 387 And generally, if tt represent any operation or series ot' operations possible when their subject is given, and then termed direct, and if, in the equation ttu = v, the subject u be not given but only the result v, then we may write u = tT^v, And the problem or inquiry contained in the inverse notation of the second member will be answered, when we have, by whatsoever process, so determined the function u as to satisfy the equation 7ru = v or tttt'V = v. By the latter equation the inverse symbol tt"^ is defined. Thus it is the office of the inverse symbol to propose a question, not to describe an operation. It is, in its primary meaning, interrogative, not directive. Suppose the given equation to be (.&^'+^'-d^-+^")"=" (!«'• Then on the above principle of notation we should have / d"" , d''-^ . \-' ^dx"" ^ dai or, with not less propriety of expression, 1 """^F 7"F^ 77^' dx^'^^dx''-''"^^ the last two equations differing in interpretation from (16), not at all as touching the relation between u and v, but only as more distinctly presenting u as the object of search. Of what avail then, it may be asked, is that analogy upon which the expression of the last two equations is founded? If a convention, it is at least a very natural one, that we should express an operation performed upon a subject, by attaching, in some way, the symbol denoting the operation to the symbol denoting the subject. The order of writing, in that family of languages to which our own belongs, has 25—2 388 SOLUTION OF LINEAR EQUATIONS [CH. XVI. doubtless determined the mode of connexion actually adopted, and wliicli is the same as if the symbol of operation were a symbol of quantity employed as a coefficient or multiplier. It comes to pass, moreover, that the formal laws of combina- tion in the direct cases investigated in Art. 2 prove to be the d same for the symbol -^ as for a coefficient or multiplier. But '^ ax ^ inverse symbols derive their meaning from the direct opera- tions to which they stand related: they are forms of inter- rogation, the answers to which are to be tested by the per- formance of the direct operations. Hence it may be inferred that the laws for the transformations of inverse expressions involving: -7- with constants will be the same as for the cor- ° dx responding forms of ordinary algebra. The analogy consists, not in the mere adoption of a common notation, but, as all true analogy does, in the similitude of relations. 4. Bolution of Linear Equations with constant Coefficients. If the equation i-^ a J w = X be given, we have but, the known general solution of the given equation being d u we see that [i-')'''=&''^^' (^7)' an arbitrary constant being introduced by the integration in the second member. If X = 0, we have {i-'^Y''='^^" (!«)• These results we shall have occasion to refer to. ART. 4.] WITH CONSTANT COEFFICIENTS. 389 Ex. Now suppose the given equation to be we have, on separating the symbols, or, by Art. 2, Hence (-; 'b]u = l-j a] X, \dx J \ax ) »=(i-^ni-«r^' (^^^- On comparing this with (19) we see that, in inverting a system composed of two operations performed in succession, the order of the operations themselves is inverted. This is evidently true whatever may be the number of successive operations, the last to be performed being always the first to be inverted. From (20) we might deduce the actual value of u by suc- cessive applications of (17). Such was the method once em- ployed. But it is better to proceed as follows. From (19) w^e have «={(i-«){i-')r^ ™- Now by the known theory of the decomposition of rational fractions {{m-a){m-- 6)]-^ ^N^{m- af' ^N^{m- h)-' . . . (22), N^, N^ being functions of a and h, which may be determined in various ways, but most directly by multiplying both sides of the equation by (m — a) (m — h), and equating coefficients. 390 SOLUTION OF LINEAE EQUATIONS [CH. XVI. Now the suggested transformation of the expression for u given in (21) is ax J \ax r^=^>(i-'*r^+^^(i-^r^-(2^> And, from the very definition of inverse forms, the proper test of the vahdity of this transformation is, that the performance of the direct operation [-^ ^)(;7 ^)on the second member shall reduce it to X. Effecting this operation, and remembering in so doing that d d -^ — aand-i b are commutative, and that by definition (tx dec I a] I -r — d] X=X, the second member becomes \dx J \dx J or {N, + F,)'^-(b^\ + aN^X (24), and this reduces to X if N, + N^ = 0, hN^-haX, = -l (25). But these equations for the determination of N^ and N^ are the same, and necessarily the same, as we should have found by multiplying, as above indicated, (22), by {m — a) [m—h), and equating coefficients. The two series of operations only differ in that ~ occupies in the one the place which m occu- pies in the other. Determining N^, N^, we see that u may be expressed in the form i ^ C? j u 'Mii-'THi-'TA «• ART. O.] WITH CONSTANT COEFFICIENTS. 391 Hence, by (17), u = -^ je"" U'''Xdx-e'''L-'^Xd(X (27), and as, on effecting the integrations, two arbitrary constants will be introduced, this is the most general value of u. 5. In like manner if there be given the general linear differential equation with constant coefficients and if we represent by a^, a^, ••• <^„ the roots, supposed all dif- ferent, of the algebraic equation m"+^,m"-' + ^2m"''--+A=^ (29), then the given equation may be expressed in the form d \(d \ (d \ ^ ,dx ^J\dx y \cix V ' whence «={(i-"')(i-»^)-(i-«»)}'^^ the decomposition in the second member formally resembling that of the rational fraction. If the equation (29) have r roots equal to a, there will exist in the resolved expression for u a series of terms of the form * This theorem was first published in the Cambridge Mathematical Jour- nal (1st series, Vol. ii. p. 114), in a memoir written by the late D. F. Gregory, then Editor of the Journal, from notes furnished by the author of this work, whose name the memoir bears. The illustrations were supplied by Mr Gregory. In mentioning these circumstances the author recalls to memory a brief but valued friendship. [See a note on page 108 of Boole's Finite Differences, I860.] 392 SOLUTION OF LINEAR EQUATIONS [CH. XVt. or, which is preferable, a single term of the form A, B,... B being determinate constants. Now since, by (17), [^ - a) X= e'''' {e""^ Xdx ; therefore (j- - a) X= (^ - a) e''" fe""^ Xda: = e"" Je"''" (€«" fe-«" X(^^) dx = e Proceeding thus, we have (i-«r^=^°'/--""^^''^^' (^^)- Ex. 1. Given J^, + 4j^ + 3?^-4|^-4y = X. \mX CLX (XX ctx This equation gives, on decomposing the complex operation performed on y, dx J \dx J \dx therefore 2' = {(i+ 2j(i + l) g" l)}"'^ TVT 1 4m+ll 1 1 (m+2)\m-^l)(jn-l) 9(m+2)' 2(m+l) ^ 18(m-l)• Therefore ART. 6.] WITH CONSTANT COEFFICIENTS. 393 Ex. 2. Given ^-^ + if^u — X. Here i^=(i— a + nM X Now K+ ^^')"'= 2w7=a) [{^-^v(-i)r-w+^v(-i)r]. 27iV(- 1) 1 Hence tc = 2nV(- 1) But e"^V(-« /'6-«^V-i)XcZaj = {cos ??a? + ;v^(— 1) sin nx] \ I cos ncc Xdx — ^{— 1) (sin nx Xdx ^cos ?ia? — /v/(— 1) sin?iic} J \cosnxXdx + ^{— 1) lsin?^aJX(iiz;^, whence, on substitution and reduction, u = -\ sin nx cos iza? X^^^c — cos nx j sin nx Xdx\ . 6. When the second member X is a rational and integral function of a?, the final integration may be avoided. For, representing the given equation in the formy (-7-J'M=X+ 0, we have «=(/(sr^M/©}""» • <»*)■ A particular value of the first term will be obtained by de- { ( d^Y^ • • d veloping ■{/(;7-)r in ascending powers (so to speak) of -7-, and then performing the differentiations on X, while the general value of the second term will introduce the requisite number of arbitrary constants. 394 SOLUTION OF LINEAR EQUATIONS [CH. XVI. Ex. Given -r-r, + n^u = 1 -\-.x -\-x^, ax + C^ cos nx + C^ sin nx = n~^ {l + x-\- x^) — 2n~^ + (7^ cos nx + C^ sin na?. The validity of the transformation of the inverse form f d^ A~^ (7-^ + n^\ by development, as of its other transformation by- decomposition, is tested by performing on the result the direct operation -^ + n^. We take occasion to notice that different transformations, while equally valid, do not of necessity con- duct us to solutions equally general, nor have we any right to expect that they should. Each solution is an answer to the question contained in the given inverse form, but that question may admit of different answers, and no solution is general which does not include them all. The final integrations may also be avoided when X consists of a series of exponentials of the form e"*"^ with coefficients which are either constants, or rational and integral functions of a;. f dY Since [ -^ J e*""" = m^e'""', we have, for all interpretable forms of/f-,- j , the relation /©^"'=/We'«' (3-5), the second member expressing the complete, because the only, value of the first member when/f -^ j is rational and integral, ART. 6.] WITH CONSTANT COEFFICIENTS. 395 but a particular value of the first member when/f-7-j is inverse, the test being as before. Hence, if the given equation be /f j- j u = S^^e"'"', we have the second term introducing the requisite number of arbitrary constants. Again, if, in any expression of the form / f-^- je'"''^, we convert t- into V" + -7^ , where -y- operates on x only as ax ax ax ax d contained in e"*^, and -~ operates on x only as contained in X, we have ^ \dxj ^ \dx dxj =/L4.).-X.by(35) Hence, dropping the suffix which is no longer necessary, since X alone follows the operative symbol, we have When therefore X is a rational and integral function of ic, a particular value of the first member may be found from the 396 SOLUTION OF LINEAR EQUATIONS [CH. XVL second, by developing the functional symbol and effecting the differentiations. And that particular value may be made general, as in the following example. Ex. Given^-3^ + 2i* = ^e"^ ~ \\dx J \dx J) [\dx J \dx J] r 1 2m -3 d I xe'^'' (2m - 3) e"'^ n x . n 2x mx = € (m - 1) (m - 2) {(in - 1) (m - 2)}"' Again, the theorem (37) relieves us from any difficulty arising from cases of failure referred to in Chap. IX. Art. 9. Ex. Given ( -^ a\ u = e"''. Here .= (^^ -«)"%- = . -(^Jl by (37) When the second member X involves terms of the form A cos mx, B sin mx, &c., we may either substitute for them their exponential values, or we may employ directly the easily demonstrated theorem . / c?^ \ sm J,, 2\ sm f -^, mx = f i—m) mx, •^ \dx J cos *^ ^ cos ART. 6.] WITH CONSTANT COEFFICIENTS. 397 d^u Ex. Given — ^ + n^u = Sa^, cos (mx + h). dx m \ I ■^®^® ^""(^'"^^y ^^-^°^^^^ + ^) + (^^ + ^V ^ -^ a cos (771^ + &) ^ ^/ . = 2,-^* — ^- — 2 — + ^1 COS ^^^ + ^ sm ?2^. n — m In this example, however, the failing case which represents itself when m = n, is most simply, though not most satis- factorily, treated by the methods of Chap. ix. Art. 11. The reduction of an integral of the v!'^ order by the fore- going theory is not devoid of elegance. We have Now let X = e^, then dX_ dx €" .,dX dd' d'X dx" -6' dd ,>dX dd =^-^'£-i)il^^'^y^37)' Proceeding thus, we have ^X=e-(|-„ + l)(|-» + 2)...|x...(38). f dX" and therefore the operation denoted l^y ( 7- ) , and the com- dy \dx. pound operation denoted by are absolutely equivalent. Hence inverting both, and ob- serving that the inversion of the latter involves the inversion of the order of its component symbols, we have 398 FORMS PURELY SYMBOLICAL. [CH. XVI. _______ d^ . .y , .sfd 1.2... (71-1) |U^ ( d \d~e + (^-V^!-^V4-^ + 3] -&c.U>'^Y 1.2 = , ^ \ ,, L""' fx^^ - (^ - 1) ^"'' fXircZa; l.z...(rt — 1; [ j j . + (" - 1^) fa - ^) ^-3 JavcZ^ . . . + Jx."-wi , the result in question. From (38) we have the theorem which is important in the transformation of differential equations. Forms purely symbolical. 7. In any system in which thought is expressed by sym- bols, the laws of combination of the symbols are determined from the study of the corresponding operations in thought. But it may be that the latter are subject to conditions of jyossihility as well as to laws when possible. And thus it may be that two systems of symbols, differing in interpretation, may agree as to their formal laws whenever they both express operations possible in thought, while at the same time there may exist combinations which really represent thought in the one but do not in the other. For instance, there exist forms of the functional S3^mboiy", for which we can attach a meaning to the expression /"(m), but cannot directly attach a meaning ART. 7.] FORMS PURELY SYMBOLICAL. S99 to the symbol /f-T-j . And the question arises: Does this difference restrict our freedom in the use of that principle which permits us to treat expressions of the formy( -y-J as if -7- were a symbol of quantity? For instance, we can attach no direct meaning to the expression e ^""fix), but if we de- velope the exponential as if -j- were quantitative, we have =/ {x + li) by Taylor's theorem. Are we then permitted, on the above principle, to make use of symbolic language ; always supposing that we can, by the continued application of the same principle, obtain a final result of interpretable form? Now all special instances point to the conclusion that this is permissible, and seem to indicate, as a general principle, that the mere processes of symbolical reasoning are independent of the conditions of their interpretation. In the few instances we may have occasion to employ, verification will be easy. We take occasion to notice that, whate^^er view maybe taken of this principle, whether it be contemplated as belonging to the realm of a priori truth, or whether it be regarded as a generalization from experience, it would be an error to regard it as in any peculiar sense a mathematical principle. It claims a place among the general relations of Thought and Language. On the principle above stated we should have =f{x + h,y + k). 400 . FOKMS PURELY SYMBOLICAL. [CH. XVI. d a h V +k rr And liere, the expression e ''^ '^^, which is without meaning in itself, is to be regarded simply as the representative of the expression ^ , d , cl 1 f, d , d\^ 1 (,d . d\\ , which has meaning. And the proper test of the validity of the symbolic equation , d d , d ^ d ^ dx dy ^ dx dy t — t fc consists in substituting for each exponential form the series it represents, and comparing the finally developed results, just as we should, by developing the exponentials, verify the algebraic equation, Jim-^'kn Jim Jen 6 =66. d d It must be noted that -y- and -r- are commutative, and ax dy combine, in all respects, like symbols of quantity. We are d not permitted to write e '^■^^ e^e'*'', because x and y- are not commutative. 8. The above principle is illustrated in the solution of the following partial differential equations. Ex. Given -^z- ci ^2 = ^ (^. y)- = 2^11^1 (^, y^:'ax)'- ^, {x, y - ax) j dy, ART. 8.] FORMS PURELY SYMBOLICAL. 401 the forms of ^^ and ^^ being given by the equations ■^ [y] and ^(^ [y] being arbitrary functions of y. If ^ [xj y) = 0, we hence find or, if we represent ^jf {y) dy by ^,{y), and — j;)^; (t/) Jy As -vjr and ;^ are arbitrary, i|r^ and Xi ^^^ ^^ too. This agrees with the result on p. 370. X. Given^ + ^, + ^3=0. u We may put this in the form y-^ -\- au = 0, where a stands d'^ d^ . for —2 + -j-2 , and integrate with respect to x, as if a were a constant quantity. Remembering that the two arbitrary con- stants of the complete integral must then be replaced by two arbitrary functions of y, z, we get the symbolical solution Developing the cosine and the sine, and replacing d^ . d'\- 4y by a new arbitrary function % (y, z), we have B. D. E. 26 / d^ d^\^ 402 GENEEALIZATION OF THE [CH. XVI. + a^X (y. ^) - 073 (I? + 5p) % (2/. ^) a^ fee- cP\^ , . r. + 1.2.3.4.5 id^ + rf?J ^ ^' "^ - ^"^^ Under tliis form/iilie solution is presented by Lagrange in the Mecamqiie Analytique, Tom. ii. p. 320. Generalisation of the foregoing theory. 9. All equations, whatsoever their nature or subject, which are expressible in the form K + ^,7r»- + Ay-^ ... + AJu = X. (1), where tt is an operative symbol subject to the laws 'jrau = airu, ir (u + v) = iru -\- irv, 7r"^7r''u — tt'""*"'' u, a being a constant and u and v functions of w, admit of trans- formations analogous to those of Art. 5, Thus, since u = {7r'' + Ay-' + A^-^ . . . + ^ J"' X, we shall have, when the roots a^,a^,.,,a^ of the auxiliary equation m" + A^nt-' + A^m""-^ . . . + ^,^ = are real and unequal, the transformation u = ^\{-K- o,)-' X+ N, (,r - «,)-X. . . +iV„ (^-o,.)-'X. . . (2) , the coefficients iV^, N^,...N^ being determined as in Art. 5. The legitimacy of this transformation is proved by operating on both sides of (2) with tt" +J^^7r''"\.. + J „, and shewing ART. 9.] FOSEGOING THEORY. 403 that (1) is reproduced with the same conditions for deter- mining iVj, JVg, . . . iV^ as if TT were a symbol of quantity. But the question of its completeness, of its conducting, through the performance of the inverse operations (tt — a^~\ &c., to the most general solution of (1), is one that we are not called upon to determine a priori. In all the cases we shall have to con- sider, its completeness will be obvious. CuU (1 }J Ex. The equation -^-{^x-^ 1) -^ + (cc^ -t- a? - 1) w = d is reducible to the form tt (tt — 1) z^ = where tt = -, x. w=(7r-l)-'0-7r-'0. Let (tt — 1)"^ = y, then, since (tt — 1) 2/ = 0, we have dv ^-^^^ In like manner, if tt ^0 = 0, we find Therefore u — c^e ^ — c^e^ . th A very interesting application of the same theory to the solution of partial differential equations is afforded by what Mr Carmichael has termed the index symbol of homogeneous functions. Cambridge and Dublin Math. Journal, Yol. vi. p. 277. Since, if u^ represent a homogeneous function of the a degree of the variables x^, x^,...x^, we have du„ d'U du„ ,^. 12 n it follows that, if we represent the symbol x^-j— ... -\-cc^^-j~ by TT, we shall have 2G— 2 404 GENERALIZATION OF THE [CH. XVI. and therefore, in accordance with the reasoning of Arts. 3 and 4, /(7r)w«=/(a)^a (A), an equation of which the second member expresses the com- plete, because the only, value of the first member when/ (tt) is rational and integral, but a particular value when the first member contains inverse factors. Hence, if we have any equation /(tt) u = X, where /(tt) is of the form tt"" + A^tt'"'^ + A^tt'''- . . . + A,„ and X is a series of homogeneous functions of the variables, suppose X=X, + X,+ ...&c., we get « = {/«}-' X +{/«)-' = ;/(^,)r X. + {fMr^. - + {/wi"o = 1/ («))"' X„ + {/(5))-'X, ... + {/Wr'O, by (A). To find the value of the last term, we proceed, as in Art. 5, to reduce it to a series of terms of the form ^. (tt — a)"*0, i being the number of roots equal to a of the equation/ (m) =0. Now it may, by an induction founded on successive applica- tions of Lagrange's method for the solution of linear partial differential equations of the first order, be shewn that {ir-a)-'0 = u^ (logoo,T'+v, (log^J"' ... +w,...(B), v^, w^ being arbitrary homogeneous functions of x„, X of the a*^ deOTee. 2' w o a> 1' To this result we may give the symmetrical form (tt - a)-' = u^r-' + v^M'-' ...+w,, L, M, &c. being logarithms of any homogeneous functions which are not of the degree 0. It remains to shew how it may be ascertained whether a proposed partial differential equation can be reduced to the form f{TT)u = X. ART. 9.] FOREGOING THEORY. 405 d_ dx. d Let us resolve each symbol j— > entering into it, into two. and let -y- represent -j— as operating on x^ only as entering i * into Uj and -y- only as entering into it. Also let X, -^ — \- x^ ^y— . . . + &c. = tt' and x. ^ — |- x,, -, \- &c. = it". dx, ^ c^^„ " " ' * ^ ^ (ia?, ^ ^a?, It is easily seen then that tt = tt' + tt''. We have therefore Ik'u = [it — it") U = 7TU ; therefore tt'Si = (tt — tt") ttw ( C). But as tt", in (G), operates on the variables only as entering into TT, which is a homogeneous function of those variables of the first degree, we may replace it by unity. . We have there- fore 7t'\o= (tt — 1) ITU. In the same way it may be shewn that 7r'% = (tt - r + 1) (tt - r + 2) ... ttic. And thus it is seen that any partial differential equation which is expressible in the form /{tt) u = X, on the hypothesis that -^— , -y- , &c. operate on the variables only as entering into u, is reducible to the form (^{tt)u = X, independently of such restriction. This reduction having been effected, the solution can be found by means of (^4) and [B), whenever the second member con- sists of one or more homogeneous functions of x^, x^, ... ^,,. ^ ^d^u ^ d\i od^u, ( du , diC\ , Ex. .^-^,+ 2.y^^ + 2,^^,-«(.^ + 2/^j +«» Here we have (tt'^ - nir ■\- n) u — x^ ^r y^ \ ^^ Therefore \tt [tt -V) —utt ■\-n\u=^c^ ^ y^ ^r oc", or {tt -n){TT- I) 2t = ic^ + 7/^ + a:', 406 GENERALIZATION OF THE [CH. XVI. whence u = {(tt - n) (tt - 1)}-^ {x"" + y' + x'] + {(tt - t?) (tt - 1)}"^ ,2 I ..2 „3 (2-«)(2-l) (3-7.)(3-l) ^t^, Vj denoting arbitrary homogeneous functions of the degree n and 1 respectively. 10. We may, by simple transformations, reduce to the above case various other classes of equations differing from the above only as to the form of tt ; e. g. the class in which ^ = ^1^1 d^ + ^2-'^2 ^ • • • + ^A 5^ ' ^^^' passing over such Special forms, we shall consider the general equationy(7r) u=X, where and each of the coefficients X^, X^, ... X^, as well as X, may be any function whatever of the independent variables. And we design to shew, first, how it may be determined whether a given equation admits of reduction to the more general form above proposed; secondly, how, then, to integrate it. Suppose the given equation of the n* order; then the symbolical form in question, should the proposed reduction be possible, will be (7r^ + A'^"-^ + -^X"'-+A)^ = -^^--' W- Now the highest differential coefficients in the given equation Avill arise solely from the symbol ir^, and the terms in which they occur will enable us to determine the form of tt. Thus, for two variables, we have {m^ ■\-xP\u = ilf ^ ^ + 2MX^ +N' ^, \ ax ay J ax dxdy ay + ax ay J ax \ ax ay J ay ART. 10.] FOEEGOING THEORY. 407 , . , , , . . d'^u dhb d\ ,, m which the terms containing -y-g , y-y- , y- 2 are the same as they would be, if, in the first member, -j- , y- were sym- bols of quantity. And this law is general for the highest differential coefficients. Again, the form of ir being determined, the values of A^, A^, ... will, whenever the proposed reduction is- possible, be found by effecting the operations implied in the first member of (4), and comparing with the first member of the equation given. Suppose the equation reduced to the form (4). Then, if the auxiliary equation m" + JX"' + A^""'--- + ^» = ^ (^) have its roots all unequal, we have a series of terms of the form (tt — a)~^X\ and each such term involves the solution of a partial differential equation of the first order of the form But, if the auxiliary equation (5) have equal roots^ partial differential equations of higher orders will present themselves. We deem it therefore important to shew how this difficulty may be avoided, or, to speak more precisely, how its solution may be made to flow from that of the corresponding case of linear differential equations with constant coefficients. Introduce a new system of independent variables y^.y^.-'-y^, d so conditioned as to give tt = y— . To prove that such a sys- tem exists, and to discover it, let us assume y^,y<^->yy,^, in succession, as subjects of the above symbolical equation, and examine whether the results are consistent. And first, assum- ing y^ as subject, we have ^.£+^41 +^^«S:=i (^)- X """"2 408 GENEKALIZATION OF THE [CH. XVI. Secondly, assuming y., representative of any of the remain- ing variables y^^y^^'-yn^ ^s subject, we have the equation X-^^-\-X-^ +X-^=^0 (7) It follows from the above that, if we integrate the auxiliary system X~X"'~~K ^^' the values of y^? 2^3> ••• Vn ^'^^^ ^® ^^^^ ^^^^ members of the integrals of that system expressed in the form 3^2 = 0^2' 2/3=«3"-y«=«« (^> And it follows from (6) that if, from the system jtrX-^tr^y' ^'°^' differing from (8) only in that it contains one additional mem- ber dy^,we deduce an additional integral equation connecting y^ with the original variables cc^, x^, ... x^^, that equation will give the value of y^. We see that the number of distinct auxiliary equations is precisely equal to the number of quan- tities to be determined, so that the scheme is a consistent one. The solution of the problem is therefore virtually dependent on the partial .differential equation (6), from the auxiliary system of which, (lO)^ it suffices to deduce n integrals, one expressing y^ in terms of x^„ x^, ... w^, the others determining y^, y^, ...y^,8iS functions o£ w^, w^,...x^^, in the forms (9). To the arbitrary constant in the value of y^ we may give any value we please. Introducing the new variables, the equation given now as- sumes the form which must be integrated as if u and y^ were the only varia- bles, an arbitrary function oiy^., y^, ... 3/,^ being introduced in the cplace of an arbitrary constant. Finally, we must restore to y^,y^, ... 2/,j their values in terms of ^j, ^3, ... ^,j. AET. 10.] FOEEGOING THEOEY. 409 Ex. Given (l-.7g + 2(l-.^)(l-.,)^^^ Here, the form of the first three terras shews that we must cl d have TT = (1 — ic^) y- + (1 — xy) y- , and the equation assumes the form To avoid the difficulty arising from the imaginary factors of TT^ 4- n^, let us assume two new variables, x and y'^ such d that we may have tt = -t- . Then by (10) dx __ dy _ 7 / 1—x 1 —xy corresponding to which we have the integral systems y-x , , //l+x\ , Hence, if we assume , , //l + a?\ , y-x we get the transformed equation therefore u = cos nx^ (y) + sin nx^lr [y)^ or, restoring to x and y their values, 410 EXERCISES. [CH. XVI. EXERCISES. dx"^ dx^ dx d7'~ dx'^ ' m = 3. Determine tlie solution of tlie above equation when _ 9 ax ax dhi „ du ^ o. -Y-o + 3 ^ + zi^ = cos mx. ax ax -J a) u = cos mx. In the above example it will be most convenient to proceed thus : ti= l—~a\ cos ??MC + ( y- - cc 1 COS mx-\- e"'^ {-j-\ 1 f d X^ "" (-m^-a^r V^"^7 cosmx + e«^(q+C2a;...+c„x'^-i). 7. Solve the equation f -^ a J u = e^ cos maj. „(^it _ d^u ^d\i f du du \ ^ CH. XVI.] EXERCISES. 411 10. Solve, by the method of Art. 10, the equation 11. The sokition of any equation of the form may be reduced to that of two linear equations of the first order. ( 412 ) [cH. XVI r. CHAPTER XYII. SYMBOLICAL METHODS, CONTINUED. 1. The classes of equations considered in the last chapter might all be gathered up into the one larger class repre- sented by TT being a symbol combining with constant quantities as if it were itself a symbol of quantity. But linear differential equations do not, except under particular conditions, admit of expression in this form. Those which are of the ordinary species involve in their general expression two symbols, x and d -J- , operating in combination on the sought and dependent variable y ; and no substituted form of such equations is general which introduces fewer than two symbols in the place d of X and -y- . We propose in this chapter to employ a trans- formation which is general, and which is adapted in a very remarkable degree to the development of general methods of solution. A somewhat fuller account of it will be found in a memoir on a General Method in Analysis {Philosophical Transactions for 1844, Part ii,). Other principles and other methods will also be noticed. The following theorems, demonstrated in Chap, xvi., will frequently recur. d li x = e^, and if -^ be represented by D, then «"g = D(i?-l)...(I>-,» + l)u (1), d while the relations connecting -7^ and e^ become ART. 1.] SYMBOLICAL METHODS, CONTINUED. 413 f{I))e'^^=f{m)e'^^ (2), f [D) e^^ u= e^^ f{D -{-m) u (3). The latter of these relations enables us to transfer the ex- ponential e™^from one side of the expression / (D) to the other, by changing D into D ± m, according as the transference is from right to left or from left to right. Thus, as another form of (3), we should have e>r^Qf{B)u=f{I)-m)&'^^u (4). It is an immediate consequence of the above theorem that every linear differential eqiiaiion which can he expressed in the form, (a^-5^+c^^..)^ + (a'^-6'^ + cV...)^~' + &c. = Z...(5), can he reduced to the symholical form, /o {D) u^f, [D) e% +/, {D) 62»« + &c. = r (6), Inhere T is a function of 6. For multiplying the given equation by cc", and assuming X — e^, the first term of the left-hand member becomes, by (1), (a + he' + ce2^ + &c.) D {D -V) ...{D - n-Vl)u, and this is reducible, by (4), to the form «Z>(i>~ 1) ... {D-n-Vl)u+h{D-l){D-2) ... {D - n) ehi + c{D-2){D-S) ... (D-n-l)6''u + &c., each term of which is of the general form (j) [D) e^u. The other terms of the first member of (5) admitting of a similar reduction, while the second member becomes a function of 6, the equation itself assumes the symbolical form (6). d\ Ex. 1. Given ^-o — n^u = 0. ax Multiplying by x^, and transforming as above, we get I){D- 1) u - n'e'^hc = 0. 41-i SYMBOLICAL METHODS, CONTINUED. [CH. XVII. dbU on Ex. 2. Given (1 + ax^) -r-^ -\-ax-T-± n\i — ^ (a?). Multiplying by x^, we have, by (1), (1 + ae^') D{D-l)u + ae^' Du ± n'e^^ u=^e^^(j> (e^). But e^^D {D-l)u={D-2){D- 8) e^^u ; and e^'Du = (D - 2) e^'u, whence, substituting, and collecting together terms like with respect to the exponentials, we have I){D-l)u+ {a {D - 2Y ± n'] e^^ u = e"'^ (e^ as the symbolical form. To return from the symbolical to the ordinary form of a differential equation, we must, by (3), transfer the exponentials to the left of each symbolic function /(i)), convert the latter into a series of factorials of the form J) {B — 1) ...[D — n-\-l), and then apply the transformation (1). Ex. 3. Given D {D -1) w + D (Z) + 1) eht = 0. We have in succession, D {D -1) u -{■ e' {D + 1) {D +2) u = 0, D{D-l)u + e'{D{D-l)'\- 4i) + 2} u = 0, ^d\ . , „d\o . du - . ^ Therefore, dividing by x, (i 11 (ill A symbolical equation which has only two terms in its first member may be termed a binomial equation ; one which has three terms a trinomial equation, and so on. We may deter- mine by inspection to which of these classes an ordinary differential equation is reducible. For multiplying it by such a power of x as to permit its expression in the form AET. 2.] SYMBOLICAL METHODS, CONTINUED. 41d where A, B, &c. are algebraic polynomials with respect to x, the number of distinct powers of x involved in those polyno- mials will determine ^the number of terms in the reduced symbolical equation. Ex. 4. Thus the equation (2/ It O U being expressed in the form {a + hx) x^ ^2 + (ex + ex^) ^ ^ + (qx') u, it is seen that its symbolical form will be trinomial, since the terms within the brackets involve x in the degrees 0, 1, and 2. [See the Bupplementary Volume, Chapter xxx. Art. 1.] Finite solution of differential equations expressed in the symbolical form. 2. If w^e affect both sides of the symbolical equation (G) with {f{D)Y\ then for f{D)-'f(D) write ^^{D) &c. and for {f{D)Y^T write U, we shall have u^^^{B)e'u^4>,{D)e^'u.., + 4>S^)e'hi^U. (7); and under this form the equation will be discussed in the fol- lowing section. Peop. I. The equation u + a,4>{D)e^u + a^<^{D) ^ (D - 1) e^^M... + aJi{B) cj> (D- 1) ... (D) ^ (D - 1) e^'u = ^ {B) e' cj, (D) e'u = {cp (i)) e^fu, and in general (D) e'Yu. So that, if we represent tlie symbol (f> (D) e^ by p, the equation in question becomes {i-ha^p + a^p\.,+ay)u=^ U; therefore w = (1 + a,p + a^ . . . + ciy)'^ U = [K (1 - q^pv + ^. (1 -^.pr ... +iX. (1 - qnPV] u. provided that q^, q^-'-q^ ^^® ^°^^® of the equation and that iVj, ^^...N^^ are of the forms i\^ = ^1 n-l Let (1 - q^py^ U— u^, (1 — q^p)'^ U= u^, and so on, then u = ]S\u^ + JS^^tt^ . . . + iV„^i„, where, in general, u^ is given by the solution of the equation u,-q,(D)e\=U (9). The solution of the general equation (8) is therefore dependent on that of the binomial equation (9). When (jb (D) is of the form D"* the equation (8) corresponds to the ordinary linear differential equation with constant co- efficients. ART. 2.] EQUATIONS IN THE SYMBOLICAL FORM. 417 2 Thus the equation u — ^ (f)_ i\ e^^i* = 0, which may be integrated by the above process, is only the symboheal form of the equation -r-r^ — ^u = (see Ex. 1) ; and its sokition, expressed in terms oi x, is v2 In like manner the equation u + jyjj{ — tn e^^^^ = has for its solution, expressed in terms of x, u= G cos qx + C" sin qx. But, when cj) {D) is not of the form D~^, the equation (8) wdll represent an ordinary equation with variable coefficients. Ex. 5. Given (a? - 2,x' + 2x') ^ + (4x - Qx') ^ + (2 + Qx) u = ax\ The symbolical form of this equation is (Z) + l) (i>+ 2) u-Z{D+l) (D-2) e'u + 2(D-2)(I)-S)e''u = a6^', whence D + 2 (D + 2)(D + -i) ^ (»+2)(» + l)' D — 2 ae^^ or, putting ^:^e« = p, ^^^K^^TI) = ^' (1-3/3 + 2^^)^^=^. 1 / 2 1 Hence w = - — :^ 7^^ T = 1 - 3/9 + 2/3^ VI - 2/0 1-/3. = ^u^-u^ (a), ■where ^«i = (1 ~ 2p)'' T, w^ = (1 " P)~' ^ • B. D. E. 27 418 FINITE SOLUTION OF DIFFEEENTIAL [CPI. XVII. From tlie former we have whence {B + 2) w, - 2 (D - 2) e^w., = -^ ; and this gives {oc — 2:c^) -^-^ + (2 + 2x) u^ = (h). In like manner we find, for ii^^ (,_,=) *l+(2+.)„^=j-;- («)• The vahies of u^ and u^, determined from (&) and {c), and substituted in (a), will give the complete solution. If a = 0, we find u = — ^-^ '—- — — . 3. We proceed to consider more fully the theory of the binomial equation u-\-{D)e'hi=U (10). Pkop. it. The equation u + cf) (D) e''^u = U will he converted into V + {D + oi) e^'^v = V, hy the relations For assume u = e'^^v, and, substituting in the original equa- tion, we have e'^^v + (/) (D) e^'^^''^^ = U; therefore e'^^v + e^'^^ (JD + n) t^'^v = TJ, by (8), Let U= e"^F; then the above becomes as was to be shewn. Thus in any binomial equation we can convert ^ ip) into (^ {D -\-n), n being any constant. AET. 8.] EQUATIONS IN THE SYMBOLICAL FOEM. 419 Peop. IIT. The equation w + ^ {D) e^ho = Uwill he converted into V + '\lr (JJ) e^^v = F, hy the relations , where P. , , -J. denotes the svmholical product ^\D){D-r)(^{D-%^)„. 'f{lJ)f{D-7')f{I)-2r)...' For, assume ii=f(I))v,Sind, substituting in tlie original equation, we have f(I))v + (D)e'J(D)v=U; therefore f{D)v + {D)f{D - r) e'^v =U,hj (4), _. ,<^{D)/(i)-r) /(-O) e^v = {fm-'U. (11). Comparing this with the equation v + i/r {D) e^h= V, we have /(i>) ^^ '' therefore /(D) =|ig)/(I>_r). Hence /(i)_,) = |^grl)/(D_2,.), and so on; wherefore the value oi f{D) will be represented by the mfinite product ^fo)f%~-r)'f{D~-tr )Z. ' ^^'^^^ (11) becomes with the relations 27—2 420 FINITE SOLUTION OF DIFFERENTIAL [CH. XVIL As this proposition is of great importance in the solution of differential equations, it will be proper to examine the conditions which its application involves. Evidently they consist in such a choice of the form of ^fr [D) as will render the svmbolical product P^-7--W,x finite, and the transformed " ^ ^Y [JJ) equation (11) integrable. That the expression of P^ j may be finite, it is suf- ficient that for every elementary factor ;)^ {D) occurring in the numerator there should correspond a similar factor y(D± ir) in the denominator, i being any integer or ; and vice versa ; for p X(D) ^ %(P) %(i>-r ) x(^-2r ) „ . 1 X{D + ir) ^ {i> + (^• - 1) r} . . . X (-^ + r) ' which is a finite expression. Again X {D) xi^)xi^-'^)"' ''x{D-ir)~x(.D-ir)x[D-(i + l)r].., = x{D)x{D-r)...x[D-{i-l)r], which is also finite ; the product of any number of such ex- pressions is finite also. Hence, if x (P) be any elementary factor of <^ {D), it may be converted into % (Z) + ir) ; for let (^D) — x (-^) %, (^}> and let i/r (D) = n^ (Z) + ir) Xi {D), wherein Xx ifi) denotes the product of the remaining factors, then ^{D)^ X(D) 'f{D) ""-xiOtir)' which is finite. Hence also, if ^ [D) involve any factor of the form —^^ ^ X(J^±iry it may be made to disappear ; for let (p (D) = ^^ . Xi (^)> ART. 4.] EQUATIONS IN THE SYMBOLICAL FORM. 421 and let if- (D) = Xi (P), then wliicli is finite. [See the Supplementary Volume, Chapter xxx. Art. 3.] 4. We see, then, that there are two distinct kinds of trans- formation to which the Proposition may be appHed. In the first kind cf) (D) is converted into another symbolic function i|r [D) without any loss of component factors, whether of nu- merator or of denominator, but only with such change as consists in the conversion of D into D ± ir. And here the order of the transformed equation is the same as that of the equation given, and, its solution introducing a sufficient num- ber of arbitrary constants, no others need to be introduced, either in the prior determination of V or in the subsequent derivation of u. But in the second species of transformation some component factor of ^ (D) (usually of the form fyj^j^ where a — & is a multiple of r) is lost, and the transformed equation being of an order lower than that of the equation given, the deficient constants of its solution must be supplied, either beforehand in the determination of F, or subsequently in the derivation of ii. If in the former, any constants, sufficient in number, introduced by the performance of \Pr%YT)\\ ^ will serve the purpose. If in the latter, all the constants introduced by the performance of F^ T7jy\ ^ must be retained, but their subsequent relation must be de- termined by means of the differential equation. [See the Supplementary Volume, Chapter xxx. Art. 4.] The reason why the constants connected with the disap- pearing factors are arbitrary in V alone, is, that V enters into no other equation than the one in whose solution those con- stants are found. If, however, the entire series of constants in Fbe retained, they will be reduced by the subsequent differentiations in passing to the value of it. 422 FINITE SOLUTION OF DIFFERENTIAL [CH. XVII. All that may seem obscure in the above statement will be made clear by the following examples. ft 11 \\ fi Ex. 6. Given -y-^ + g'^i^ ^ = 0, an equation occurring in the theory of the earth's figure. The symbolical form is Now we may, by Prop. Ill, directly reduce this equation to the form which, by Prop. I, is resolvable into two equations of the lirst order. But it is better to assume as the transformed equation ^+i)(fc)^""=o (^)' the solution of which is known already. Art. 2. By Prop. II, assuming u = 6~^^zu, we have Again, by Prop, iii, we can pass from (c) to (6) by assuming Hence u== e-^^{D -1) (D -S) v = —Jx^ -,—„ — Sa)-f-+ S]c sin (go) + c) o) \ dx^ dx ) ^^ ^ on restoring x and putting for v its value in terms of x. ART. 5.] EQUATIONS 11^ THE SYMBOLICAL FORM. 423 Effecting the differentiations, we find '^^ " ^ |(^' ~ ^ y ^^^ fe^ + ^') ~^cos (qx + c)^ [d). We might have proceeded directly from (a) to (b) by Prop. Ill ; but, had we done so, the final reductions would not have depended on differentiations alone. Thus we should have had ^ D (D-l) D-1 {D + 2){D-S) D + 2 = {1 - 3 (D + 2)-^l v=(l- 3e-2^i)-V^) V' whence, restoring x and giving to v its previous value, we should be led to the same solution as before. 5. The two forms of solution above presented illustrate an important observation, viz. that when in the transition from ), by Prop. Iii, the reductions consist in aug- menting, if we may be allowed the expression, D in factors of the denominator of (p {D), or in diminishing D in factors of the numerator, they will be effected by differentiations ; while those reductions which consist in augmenting D in factors of the numerator of ^ (!>), or diminishing it in factors of the denominator, involve integrations. And it is one use of Prop. II, that it enables us, in many cases, so to prepare the given symbolical equation that the final reductions shall depend on differentiations. Ex. 7. It is required to determine the symbolical form and character of those differential equations of the n^^ order, the solution of which depends on that of the equation The symbolical form of this equation is ^^D{D-l)...{D-n-\-l) ^ ^' where V is the symbolical form of fyj X, i.e. the result 424 FINITE SOLUTION OF DIFFERENTIAL [CH. XVIL obtained by writing e^ for x in the n^^ integral of Xdo^, no constants being added in the integration. From inspection of [a], it is evident that the class of equations sought must, on assuming x — e^, be expressible in the form in which we shall suppose the quantities a^, a^... a^Xo be ranged in the order of decreasing magnitude. Put u = e'^^^Mj, then by Prop, ir, The first factor of the denominator of (^ (IT) in (c) now agrees with the first factor in that of -v/^ [B) in (a). In any of the remaining factors we may, by Prop. Ill, convert D into D + in, i being any integer, — hence, that they may all cor- respond with tlie factors of i/r (D), it is necessary that each of the quantities ^2 ~ <^i + 1 0^3 — «j + 2 a^ — a^ + S a^—a^-^-n — \ . ,. n n n n ^ ^ should be equal to a negative integer or to 0. And in this statement the conditions of finite solution are involved. The value of u will be deduced from that of v by differen- tiation, for since a,^— a^< — 1, and so on for the remaining factors to which P^^ is to be applied, Ex. 8. Given -p^ ^^ — r, — - u + o^u — 0, where i is an integer. This equation, which includes that of Ex. 6, presents itself in various physical problems (Poisson, Theorie Mathematique de la Ghaleur, p. 158. Mossotti, on Molecidar Action, So.). ART. 5.] EQUATIONS IN THE SYMBOLICAL FOKM. 425 Its symbolical form is Hence, by the last example, = e-^'{D-l){n-S),..{D-2i + l)v (h), cVv wliere v is given by -^-3 + c^^ = 0. The expression (h) may be reduced to a more convenient form, as follows. Since /(D -a) = e^^f(D) e'^^ we have V COX/ Hence, according as the npper or lower sign is taken in the original equation, we have _ 1 / 3 <^ Y Cj cos qx + Cg sin ox , . ''~^r dx) ^^ ^ ^''^' ^=^^r dx) ^- -(^^^ Ex. 9. Given -;— . - a^ -7-^ ^^ — ^-^ = 0. aa; a?/ x Comparing this equation with the last, we see that its solution may be derived from (d) by changing therein q into a -J- , and c^, c^ into arbitrary functions of y following the exjDonentials. Hence we shall have 426 FINITE SOLUTIOX OF DIFFERENTIAL [CH. XVII. U = -^xr Od d d ax-r . , . —ax ■ 1 / 3 dVe ''cp(y)+e '^f{y) 3 dV (p(y + ax) + '^{y — ax) af^' K^ dx) ^ x"^' • The reason why the arbitrary function ^ (?/) must be placed d_ after e '^^ and not before it, is that, in the derivation of the exemplar form, the arbitrary constant takes its place after, and not before e^"". Here indeed we may transpose the constant, but when q is converted into a -j- we have dy and here the arbitrary function cannot b& transposed, since t/ and -r- are not commutative. •^ di/ The principle here illustrated, and which is a very im- portant one, is that all conclusions founded on community of formal laws should stop short of interpretation. The form should be kept distinct from the matter. There is perfect analogy between the theorems {i-^T'=^"&^-"'' and (^-a^r0^e<(pri<0, \dx ay) \dxj but not between the theorems because in the formation of the latter interpretation has been employed. ART. 5.] EQUATIONS IX THE SYMBOLICAL FORM. 427 The above example is one in which Monge's method of solution would fail, except for the particular case of i = 0. And this gives occasion to the remark that symbolical methods are not, as they have sometimes been supposed to be, valuable only as abbreviating the processes of analysis. There are in- numerable cases in which they afford the only proper mode of procedure. Ex. 10. Given This equation occurs in some researches of Poisson on definite integrals. The symbolical form is ^ (Z) + 2.-2)(D+2n-2-p) _ This equation is integrable in several distinct cases, but we shall examine here the particular case in which n is an integer. Assuming as the transformed equation, V- J, ^6^^V= V (b), it being necessary to introduce V because the transformed equation is of an order lower than that of the equation given, we have, „ Z) + 2n - 2 u=F,, ^ V = (D + 2?2-2)(Z> + 2n-4)...(Z) + 2)?; (c), = (D + 271 - 2) (D + 2^-4) ... (Z> + 2) F: The latter equation gives for V the general value, of which it suffices to retain one term. Retaining the first, 428 FINITE SOLUTION OF DIFFEEENTIAL [CH. XVII. substituting in (h), and operating on both, sides with D+p, we get -29 {D-\-p)v-{D + 2n-2-p) e^^v = c^{p-2) e Restoring cc, and integrating, a value of v is found, involv- ing two arbitrary constants, whence u will be given by u = (^l + 2„-2)(.i^ + 2„-4)...(4+2)....(^). The proposed equation is also integrable when p> is an odd integer, and when 2n — j9 is an even integer. In the former case we may assume as the transformed equation, {D-\-p)[D-\-p-l) which must be integrated by Prop. I. In the latter case we must assiune V — e'^^v= V; ' but in this case two constants must be retained in V; viz. one from each set of the reducing operations by which the factors of

-l) "^ = ^- If we apply Prop. II. so as to convert D into D — m, and then by Prop. III. reduce the equation to the general form (13), we shall obtain the final solution in the form 430 FINITE SOLUTION OF DIFFERENTIAL [CH. XYII. 7. Pfaff's Equation. The differential equation, (a + 6^") x'' ^^2 + (c + ex^) x ^ + (f+gx'') u^X. (a), which inducles all binomial equations of the second order, has been discussed by Euler, and, with greater generality, by Pfaff {Bisquisitiones Analyticce). We propose to investigate the conditions under which it admits of finite sokition. It suffices for this purpose to consider the case in which X=0. The symbolical form is then h(D-n){D-n-^l)+e(D-n)+g _ If n is not equal to 2, it is convCDient to change the inde- pendent variable by assuming n6 = 26', whence d _n d d~e~2dO" « So that changing nO into 26' , we must change D into ^ D. The result may be expressed in the form, "+-S^^w#^^"'"=o ••>)' Vv^here a^ and a^ are roots of the equation, 6_(f-«)(f-.-l)+.(f-»)+^ = id), and ySj, /S., are roots of the equation, 2A"2""V^'"2"+*^=^^ ('^^- 1st, By Prop, ill, (c) can be immediately reduced to the form 5 (J-«.)(7)-.,-l) " 2 ART. 7.] EQUATIONS IN THE SYMBOLICAL FCEM. 431 and then resolved into two equations of the first order, if we have at the same time a^ - c/g, and j3^ - ^^ odd integers. 2ndty, The equation can, by Prop, iii, be reduced to an equation of the first order if any one of the four quantities «i-A> ^i-ft. «2-ft' «2-^2 is an even integer. Srdly, It is easily shewn that by Props, ii. and iii. (c) is reducible to the integrable form (13) if the quantities P1-P2 and «i + a2-/5x-^2 are both odd integers. 4thly, It is in like manner reducible to (14) if the quan- tities a^ - a^ and ot^ + a^ - ^^ _ ^^ are both odd integers. These results may be collected into the following theorem. The equation (c) is integrable in finite terms, 1st, if any one of the four quantities represented by a — ^8 is an even integer; 2ndly, if any two of the quantities «l-«2' ^l-.^2' ai + ^2-ft-/^2 are odd integers. In this theorem the integral values are supposed to be either positive or negative, and the even ones to include the value 0. The above results are equivalent to those of Pfaff, as pre- sented with some slight increase of generality in a memoir by Sauer [Grelle, YoL 11. p. 93). Pfaff's conditions are how- ever exhibited in so complex a form as to render the com- parison difficult. His method, it is needless to say, is wholly different from the above. [See the Supplementary Volume, Chapter XXX. Art. 5.] 432 SYMBOLICAL EQUATIONS [CH. XVII. Symbolical equations which are not binomial, 8. Altliougli processes of greater or less generality may be established for the treatment of equations which, when symbolically expressed, involve more than two terms in the first member, yet their reduction if possible by some preli- minary transformation to the binomial form should always be our first object. We purpose here to illustrate this obser- vation. Ex.12. Given ^ = a- ^^ dx^ (2cx — x'f Writing this equation in the form (2c -xyx^^^-\-by^a {2cx - x^, we see at once that its symbolical form will not be binomial. Assuming y = (2c — x^'u, we have on reduction . .2 ^^^ Q 2 ^^^ '^^ fm — 1) a?^ + & _ aoi? Now let m be so determined as to make the numerator of the third term divisible by its denominator. This involves the condition m(m-l)+^ = (a), while the differential equation becomes rPqi rJrti aOu {2c-x)x'~^,-2mx'^-m{m^l){2c + x)u=j^^--^,, of which the symbolical form is (D-m)iI)+m-l)u-^^(D + m-l){D+m-2)eOu=j^^^^^—^„ whence, operating on both sides with (D +m — 1)~^, {D-m)u-^~{D + m-2) eht = ^ e), we write -v/r {e'^V(-i)| and ^|64>V{-i)}, as we are evidently permitted to do, and if we observe that generally ART. 8.] WHICH ARE NOT BINOMIAL. 43 i =/ [e{'^+ios^^V(-i)}V(-i)] =y {e.|.V(-i)-log^j /( =/~^ (/). we shall ultimately find + ^-"')-x(1^')} (.^). which is the complete integral. For a discussion of this result, and for the finite expression for Laplace's functions to which it leads, the reader is referred to a paper on the Equation of Laplace's functions in the Cambridge Mathematical Journal. (New Series, Yol. i. p. 10. "i If in the equation (a) we make the third instead of the fourth term to vanish, which gives for r the values - and 2 ^r- > ^^^ *^^^ assume -- J^ = t, we shall obtain, taking the second value of r, the symbolical equation ^+ B{IJ-1) ' ^~^- Now by Propositions ii. and iii. this is reducible to the inte- grable form .2 "'^ i>(Z>-l) "" = ^' by the relation 486 SYMBOLICAL EQUATIONS NOT BINOMIAL. [CH. XVII. V = 6-("^')^ D{D-l).,.{I)-n)w Hence we find wlience u is known. Let us examine the form of the solution, when, as is com- mon in the expression of Laplace's equation, we replace /i by cos 6. We find ^ = eot^, | = -sin^^^, whence 1 1- ^J(1 + f) = cot 1 6. Substituting, and observing that u = (sin ^)""'V, we have u = (sin 5)-"- (sin^e i)""f (°°t f)°+ «^ (*^^|)} • And hence, restoring to a its meaning, introducing arbitrary functions for constants, and effecting one of the differentia- tions, we may deduce the following solution of Laplace's equation, viz. : u = (sin ey (sin e ^ sin ey F^ |e^V(-i) tan || ^ (16). + i^ Je-^V(-i)tan^ Under this singularly elegant form the solution, obtained by a different method, was given by Professor Donkin. (Fhilo- soijhical Transactions, for 1857.) ART. 9.] SOLUTION OF LINEAR EQUATIONS BY SERIES. 437 Solution of linear equations hy series, 9. Prop. IV. If a linear differential equation ivhose second member is he reduced to the symbolical form /„ (B) u+f, {D) 6% +/, (D) 6% ...+/. (2)) 6««« = ... (17) {Art. 1), then a particular solution will be u=XiK/-' (18), the value of the index m in the first term being any roof of the equation f (m) = 0, the corresponding value of u^^ an arbitrary constant, and the law of the succeeding constants being expressed by the equation For the form of u assigned in (18) will constitute a solution of (17) if, on substituting that form for u in the first member of (17) and arranging the result in ascending powers of e^ each coefficient should vanish. And this, as we shall see, will take place if the coefficients are subject to the relation expressed by (19). Assuming then u = Sw^e™^, we find, /„(D) « = 2/„(i))»..e™» = 2/.(m)«,„6»«, by (2), and so on. In the first of these, we see that the coefficient of any particular term e™^ is f^ (m) u,^. In the second, the co- efficient of 6^"*+^'^ isf(m + l)Umy ^^^ therefore the coefficient of e™^ is f{m)u^,_^. In the third, the coefficient of e™^ is f im) w„j_2 ; and so on. Thus the aggregate coefficient of e™^ is /o (^) ^m -^-/l (^^) ^m-l +/2 (^^) ^m-2 •••+/« (^0 ^m-n > and this,. equated to 0, expresses the law (19). 438 SOLUTION OF LINEAR [CH. XVII. Let u^ + e2^i^ = (a). Now the equation (D^ + w^) w = gives i^ = ^ cosnd +B sin w^ (6), substituting which in (a), and equating to the coejSicients of cos nd and sin n6 in the result, we have D'A + 2nDB + e^^J. = 0, D'B - 2nDA + e^'B = 0, whence A = Sa^^e™-^, B =-^h^/^^ , with the relations, mV^ + 2mn5^ + a^^_2 = 0, m'^^ - 2mna,^ + 6,,_, = 0, AET. 11.] EQUATIONS BY SERIES. 44<1 and therefore, Thus the solution assumes the form, u — cos {n log x) [a^ + a^cc^ + ajic^ + &c.) + sin {n log ^) (/?q + h^x^ + J^cc* + &c.), wherein a^ and ^^ are arbitrary, and the succeeding coefficients determined by (c). The fundamental equation (19), written in a reversed order, determines the law of the formation of the coefficients in those solutions of (17) which are expressible in descending powers of x. The number of such solutions will be equal to the degree of the equation f^ (in) = 0, but their respective first exponents will be its roots severally diminished by n. For the extension of the above theory to the case in which the given differential equation has a second member X, the reader is referred to the original memoir. Theory of Series. 11. The relations which enable us to express the integrals of differential equations in series, enable us also to reduce the . summation of series to the solution of differential equations. Thus, from Proposition IV. it appears that if u = Xu^x"^, where the law of formation of the successive coefficients, is /o (^^) ^^ +/x (^^) ^^^-l • • • +/« (^) ^m-« =0 (23), the value of u will be obtained by the solution of the differ- ential equation /, {D) u +X (2>) 6«» . . . +/. (D)6"«« = (24). "We suppose here /^ {m),f^ (m) .../„ (m), to be polynomials, and that the series is complete ; i. e. contains all the terms which can be formed in subjection to its law expressed by (23), the first exponent being therefore a root oi fiin) =0. 44^2 THEORY OF SERIES. [CH. XVII. When the series is incomplete, the first member of the differ- ential equation will be the same as for the complete series, while the second member will be formed by substituting in the first member, in the place of u, the series which it repre- sents. It is obvious that all the terms will disappear, except a few derived from that end of the series where the defect of completeness exists, so that the second member of the differen- tial equation will be finite. • Ex. 17. Let ,2 1.2'' +1.2.3.4'^ 1.2.3.4.5.6 ^'• Here u — ^u^^x^, with the relation, ^'"~ m(m-l) ^''"-2* Or, m{m-V)u^-{{m- 2.)' - n^\u^_^ = 0, and we observe that the series is complete, the first index being a root of m {ra — 1) = 0. Hence, the differential equation will be J){I)-\)u-{{J)- 2f - n'] ^'u^O, of which the solution, expressed in terms of x, is u = c^ cos [n sin~^^) + c^ sin {n sin~^a;). The constants must be determined by comparison with the original series. We thus find c^ = 1, Cg = 0. The following is a species of application which is of frequent use in the theory of probabilities. Ex. 18. The series p |i + ag + — YT2~ ^ '" 1 .2... 6 ^ ^' AE,T. 11.] THEORY OF SERIES. 443 occurs as the expression of the probahility that an event whose probability of occurrence in a single trial is jpy and of failure q, will occur at least a times in a + 6 trials. Kepresenting the series within the brackets by u^ and assuming q = e^, we have u = Xu^e"^^, where mu^^ -(jn+a-1) u^^_^ = 0. Hence, we shall have -n, /T-i -,\ a a (a + 1) ... (a-]-h) ,.,.^a Du~ (D + a-1) ehi = ^ — —^ \ ' e(6+i)0 1 . z ... or, restoring q, dn a a la 4- 1) ... (a + h) q^ u — — ^ dq 1 — q 1.2... 6 1—2'* Integrating which, we have /-I \ nf/^ a (a + 1) . . . (a + 5) r^ , .^ . . -, . u=={l-q)-[G ^-;2-.-^ ]9' (1 - ±|±^JV(l - ,)- d,, or, as it may be otherwise expressed, Probability = jjJ,-^3—g^ (5). The peculiar advantage of this form of expression is that, precisely in those cases in which the series becomes unmanage- able from the largeness of a and h, the integrals admit, as Laplace has shewn, of a rapid approximation {Theorie Ana- lytique des ProhahiliUs). Ex. 19. The function (1 — 2v cos co -f z^^)~" being expanded in a series of the form A^ + 2 {A^ cos co -\- A^ cos 2co ... + &c.'j, it is required to determine A^. We have (1 - 2v cos CO + vy = {1 - v€-V(-i)}-- X {1 - ^6-"V(-i))-«. Expanding each factor, and seeking the common coefficient of ertuv(-i) and e~^"V(~i^ in the product, we find, putting ^ = z/^, where generally, m {m -\- r)u^^— {m -{■ n — 1) {m ■\-n-\-T — l) u^^_^ = 0, , ., n(n+l) ... (n -\- r — 1) while u^ = -^ 7 — 7^ . 1 .2 ... r Hence the differential equation will be I){n + r)u-(D + n-l){D + n + r-l) e^u = 0, or, (D + n-l){D-^n + r-l) * Now this can, by Prop. Ill, be reduced to the form, V — e^v = V, ART. 12.] THEORY OF SERIES. 445 by the relations, u= {D + n-l)...{B^-l)[D + n-\-r-l)...{D-\-r-^l)v, Y=[{D-\-n-l).,.{D + l){D + n + r-l) ...{D + r+l)]-'U. In determining V from the latter equation, it suffices to in- troduce two arbitrary constants, one from each of the two sets of inverse operations. The final solution, in the obtaining of which the only difficulty consists in the reductions, is 12. When, in the series Xu^x"^, the coefficient u is a rational function of m invariable in form, the summation is most readily effected in the following manner. Let the series be X(f> [m) a?*"; then putting a? = 6^, u = ^^ (m) e^^ = tcf) {D) 6«^^ ^{D)te^' (25). Hence, if the summation is from m = to m = infinity, we have ^^ = 0(^)1^77; but if the summation is from m = a to m = 6 inclusive. u = cl^(D) 1-6^ Ex.20. Let. = ^+^4^ + 3^^+&c. Here (m) = therefore u = m {m — 1) (m — 2) ' ^(^fiy(^(^'' + e- + &c.) 446 GENERALIZATION OF THE [CH. XVII. The final result is u =x-i-S-^^+l^')^°°(^-^)- I Generalization of the foregoing theory. 18. As Propositions i, ir, iii. are founded solely on the .J particular law of combination of the symbols D and e^, ex- " pressed by the equation f{D) e^^u = €^Y (Z) + m) k, they remain true for any symbols tt and p, whatever their interpretation, which combine according to the same formal law; viz. f(7r)p"'u = p-f{7r + m)u (26). Thus, supposing the law obeyed, the symbolical equation, u + {D) e\ p = (/) (D) e^ the law will still be obeyed. And the importance of the remark consists in this, that an equation which, when ex- pressed by means of the sj^mbols D and e^, is not a binomial, may assume the binomial form for some other determination of IT and p. ART. 13.] FOREGOING THEORY. 44^7 If in (26), we make m = 1, we have/ (tt) pu = pf{7r + 1) ii, which shews that p may be transferred from the right to the left of /(tt), if we, so to speak, add to tt the constant incre- ment 1. This then suggests the more general law, f(7r)pu = pf(7r-\-A7r)u (29), where Att represents any constant quantity regarded as an increment of tt. In connexion with this theory, the following proposition is important. Prop. Supposing f (x) to i^epresent a function which admits of expansion in ascending ptositive and integral powers of x, it is required to dev elope f {ir + p) in ascending poivers of p, TT and p being syr)ihols ivhich combine in subjection to the law (29). By successive applications of (29) we have, m being a positive integer, f{7T)p'^u=^p''f{ir + mAiT)u (80), of which another form is p^fiir) u =f(7r — 7nAir) p^u. Again, since/ (tt + p) is, by hypothesis, expressible in a series of the form A, + ^, (tt 4- p) + A^ {'tt+pY + &c., we shall have (,r + p)/(7r + p)=/(7r + p)(7r + p) (31), for either member becomes, on substituting for /(tt -f p) the above form, A {'^ + p) + A (^ + io)'+ &c. Now, let the form of the unknown and sought expansion of /(tt + p) in ascending powers of p, be /(■^ + P) =/. W +/: W p +f, « P'+ &0 (82), = 2A {^) P'. the subject u being understood though not expressed. Then, by (31), (tt + p) If^^ (tt) p- = tf^ (tt) p- (tt + p). 448 GENEEALIZATION OF THE [CH. XVII. But (^ + p) y, (,r) p" = 27r/,„ (tt) p- + 2py;„ (^) p- = Stt/; (tt) p" + 2/„ (tt - A,7) p^*\ in whicli the coefficient of p"^ is '^/..W+A-.C'r-ATr). (33). Again, 2/„ ('^) z'" ('^ + p) = 2/„ (tt) p-TT + 2/„ (tt) p-" in which the aggregate coefficient of p"* is A WC^-^^Att) +/,,_, (tt). Equating this with (33), we have whence ^ , , 1 /• , (tt) - /• r-TT - Att) ■'"^ ' m Att ^iA/:»-.w (84), m Att if we define A/(7r), not, as is usual, by /(tt + Att) — /(tt), but %/W "/("^ ~'^'^)- The above equation determines the law of derivation of the coefficients /^ {tt),/^ (tt), &c. It only remains to determine f^ (tt). That /(> (tt) =/ (tt) may be shewn by induction from the particular cases in which /(tt + p) = tt + p, (tt + p)', &c. or, with more formal propriety, thus : Let Pj = np, where ?i is a constant, fM Pi ^/M ^¥ = "/(t^) P == npf(7r — Att) = Pifi'^-^'^)' .2 ART. 13.] FOREGOING THEORY. 449 Comparing the first and last members, we see that tt and /)^ combine according to the same law as tt and p. Thus we have / (t + ft) =/„ W +/ (tt) p, +/, (tt) p,' + &c., /o W»/i W' ^^- ^eing the same as in (32). Or, f(w + np) =/„ (tt) +/, (tt) np +/, (tt) ny + &c. ; so that, making n = 0, we have/^ (tt) =f(7r). Determining then the successive coefficients by (34), we have finally, + 172773 W^^^' ^''^' wherein it is to be remembered, that A/(7r) _ /(7r)-/(7r-A7r) Att ~ Att When A7r= 0, the symbols tt and p become commutative, and (35) assumes the form of Taylor's theorem. As a particular application of the above, suppose that we liave given the trinomial equation {D'' + aD + h)u+{GD + e)e^uA-fe''u=^0 (a), and that we desire to ascertain whether this can be trans- formed into a binomial equation by assuming IT = D — me^, p = €^, assumptions which satisfy the law /(7r)p=p/(7r + l). Here we have D = 7r -\- mp, whence f{D) =f{7r) + -^ mp+^ ^^ my + &c., B. D. E. 29 450 LAPLACE'S TRANSFORMATION OF [CH. XVII. where Att = 1, and Hence D^ -f- ai> + 6 = tt^ + a7r + 6 + (27r — l+a)mp + iii^p^, cl) + e — C7r-\- e + cmp. Thus (a) becomes {tt^ + aTT + 6 + (Stt -1 +a)mp+ rri^p^} u + (cTT + e + cmp) pu -Vfp^u = 0, or 7r^+a7r4- 6 + {(2m+c)7r+m(a— l) + e}p+(m^ + cm4-/)p^=0, and this reduces to a binomial equation, 1st, if m be a root of tlie quadratic equation m^ + cm, -\-f= ; 2ndly, if it be possible to satisfy simultaneously the equations 2m + c = 0, m (a — 1) + e = 0, equations which imply the condition 2e - c (a - 1) = 0. The discussion of the binomial equation when obtained in- volves no difficulty. For a discussion of the general trinomial equation of the second degree, the reader is referred to the original Memoir. Laplace s trayisformation of partial differential equations, 14. Laplace has developed a method for the reduction of the partial differential equation Er + Bs-\- Tt-tFp+Qq + Zz= U (36), ii, 8, T,...U being functions of x and y, which is deserving of attention from its great generality. One of the auxiliary equations in Monge's method is Bd/-Sdxd7/-{-Tdx'=:-0, AET. 14.] PARTIAL DIFFERENTIAL EQUATIONS. 451 Let two integrals of this equation be and assume two new variables, ^ and rj, connected with x and y by the equations The student will have no difficulty in proving that the given equation will assume the form ^'^ j^L^+M~-{-Nz=V. (37), d^ drj d^ dj] L, M, N, F being functions of ^ and tj. The theory of the reduction of this equation is then contained in the following propositions : 1st, The equation (37) may be presented in the form Hence, if the condition rJT N-MI-^=0 (39) be satisfied, and we assume l-j- -\- L\z=^z, we shall have The solution of the given equation is then dependent on that of two partial differential equations of the first order. 2ndly, Inverting the order of the symbolic factors, the equation is also solvable if we have dU ]Sr-LM-^=0 (40). dr] 3rdly, The equation (37) can be transformed into a series of other equations of the same form, and therefore integrated, if, for any of those equations, the condition (39) or (40) is satisfied. 29—2 452 Laplace's transformation of [ch. xvii. For, expressing it in the form (38), let, as before, (1+^)^=^' (")• Then (| + i/)/+(i^_il/-|).= F, whence z — which is of the form -%-Mz ^V A, By C being functions of f and tj. Substituting this ex- pression for z in (41), we have a result of the form ^^ +r% + M'^-]-N'z=:r ,.(42). c?f drj d^ dr) Thus the form (37) is reproduced, but with changed coeffi- cients. Hence the equation is integrable if either of the fol- lowing conditions is satisfied, viz. N'-L'M'-^=0, N'-L'M-^ = (43). a^ drj If neither be satisfied, the process of transformation may be indefinitely repeated, and should an equation be obtained in which either of the relations (43) is satisfied, the solution may be found. It has indeed been asserted that " if the given equation be integrable, we shall finally get an equation in which this essential condition is satisfied" (Peacock's Exam- ples, p. 464). The state of our knowledge of the conditions of fiuite integration does not however warrant this confidence. A discussion of the equation dz „dz ^^^ 1 7, ^^^ 4. ^ I ^-^ dy gz _ T\jf ( \ ^ dx^ dxdy ay"" hx -\- ky (Jix-\-kyY~ '"^^^ ART. 14.] PARTIAL DIFFERENTIAL EQUATIONS. 453 by Laplace's method is given in Lacroix (Tom. Ii. pp. 611 — 614), but it is far too long and too complex to find a place here. The best mode of treating the equation is probably the following. Let s and t be two new variables connected with X and y by the linear relations hx + ky = s, y ■{■ mx == ^, of which one is suggested by the form of the given equation, while the other is adopted in order to put us in possession of a disposable constant m. Transforming, and making in the result s = €^, we obtain the symbolical equation {AD[D-l)-^ED-Vg]z+j^{B{D-l)+F] e'z+ C~e''z==6''M. ..(h), in which A=ah^ + hhk + ck\ B= 2ahm + h{h + km) + 2ck, G = am^ + hm + c, E=eh +fk, F=em +/. The equation will be a binomial one, if m be determined so as to make (7=0. We have then am^ + bm + c = 0, while the symbolical equation (h) becomes and is integrable if the following condition is satisfied, viz. B--F A-E+^/{{A-E)'-4^g} — =i ~ c^ A ^-^ = an mteo^er or 0. B 2A ^ . This condition will be found to include the one to which Laplace's method leads. At the same time it is seen that the equation (h) assumes the binomial form under other conditions than the above; e. g. if we have simultaneously ^ = 0, i^=0, from which, by elimination of m, we find f(2ah + bk)-e(bh+2ck) = 0. 454 MISCELLANEOUS NOTICES. [CH. XVIL This condition being satisfied, and m determined, the sym- bolical equation becomes and is integrable if the two roots of the equation Am (m — 1) + Em + g = differ by an odd integer. There are probably other cases dependent on the reduction of Art. (13). In one respect Laplace's transformation possesses a gene- rality superior to that of all others. For its tentative applica- tion fewer restrictions on the coefficients of the given equation are necessary. But, that the application may succeed, other conditions seem to be demanded which render the estimation of the true measure of its generality difficult. And, in par- ticular instances, it is seen that it is less general than the method of the foregoing sections. Miscellaneous Notices. 15. Of special additions to the theory of the solution of differential equations by symbolical methods, the following may be noticed. 1st, Professor Donkin has shewn that, \i f{x) be any func- tion capable of development in powers of x, then whatever may be the interpretations of the symbols tt and p, we have f{p-'7rp)u = p-yi7r)pu (44). This is evident from the consideration of such cases as the following : (p~V/))^ = p'Wpp'^TTp = p~Vp, (p-vpr = p-v- (p-'r = p-v> We are thus enabled to generalize many important theorems. Thus, since \j + ^'(^)| ^ = e-^(^> ^ e<^(^)w, we have (Cambridge Mathematical Journal, 2nd Series, Yol. V. p. 10.) ART. 15.] MISCELLANEOUS NOTICES. 455 d 2ndly, Mr Hargreave, observing that the symbols -7- and d — X are connected by the same laws as x and -^ (the proof of this will afford an exercise for the student), has remarked that if in any differential equation and its symbolic solution we d d . change x into -y- , and -7- into — x, we shall obtain another form accompanied by its symbolic solution. {Philosophical Transactions for 1848, Part I.) Applying this law of duality to the known solution of the linear differential equation of the first order, it is easy to shew^ that the equation has for its symbolic solution, 2i = {^(i))r^6^(^)^-^e-x(i»X (46), where %(^)=/|x§^^' a form which had before been established on other grounds [Philosophical Magazine, Feb. 1847). Many other illustrations of the same law will be found in the memoir of Mr Hargreave referred to. Srdly, The method by which the development of/(7r + p) is obtained in Art. 13, leads to other and similar results, of which the following is among the most interesting, viz. the coefficients of the expansion in the second member follow- ing the law of Taylor's theorem, and the fuQction F{x) being equal to e^^'^"'^ /i^)- (Cambridge Mathematical Journal, 1st Series, Yol. iv. p. 214.) The last theorem enables us to integrate at once any equa- tion of the form F (..) u + F' (x) J + -^^ F" (x) £j + &c. = X, 456 MISCELLANEOUS NOTICES. [CH. XVII. where Fix) is a rational and integral function of x. For let an expression always finite under the conditions supposed. Then the given equation assumes the form f{7r)u=X, where tt = a; + -7- , and may be treated by the method of the last section. Other examples of the expansion of functions whose symbols are non-commutative — some of them admitting of a similar application — will be found in the memoir of Professor Donkin above referred to, and in an interesting memoir by Mr Bron- win {Cambridge Mathematical Journal, Vol. iii. p. 36). 4thly, Many important partial differential equations of the second order admit of reduction to the form du dv du dv _^ dx dy dy dx ' whence an integral u=f(v) may be deduced. Thus the equation /d^d^ __d4&^\ /^^_^2^_ # ^_ f# + ^i") 5 _ #: \dp dq dq dpj^ ^ dp \dq dp J dq where (f> and yfr represent any given functions of jp and q, may be expressed in the form d{(j)-x)d('ylr-y) d {(j) - x) d {yjr-y) ^^^ dx dy dy dx ' whence (j) — x = F{'^ — y) is a first integral. Mainardi has shewn that nearly all the equations which occur in Monge's Application de V Analyse d la Geometrie, admit either of the above reduction, or of a purely symbolical mode of solution. {Tortolini, Vol. v. p. 161.) 5thly, The Author is indebted to Mr Spottiswoode of Oxford for an interesting communication on the laws of combination of symbols which are at the same time linear with respect ^ + 1 = 0, CH. XVII.] EXERCISES. 457 to -r- , -T- , &c. and linear with respect to x, y, &c. The ax ay following is one of the results. If, assuming d d d ^ d a partial differential equation can be presented in the form /K, 7^2)^ = d d on the assumption that -j- and -^ operate only on the subject u, then it can be expressed in the form i^(7r^, ir^u = 0, indepen- dently of such restrictive hypothesis. It might be added, that all such equations are reducible to equations with constant coefficients, by assuming 1 / 2 . 2n1 r ^ fX+y\^ \og(x' + yy = co, log(^^-3^j =y. To the above might be added many other special deductions, isolated now, but destined perhaps, at some future time, to be embraced in the unity of a larger theory. EXERCISES. o ij du 1. Integrate x^ -^^ '^^^li (fo^u = 0. 2. Integrate (a?^ - a?') 7^ - (^ + 3a;') ^ + (1 - aj) it = 0. 3. Riccati's equation is reducible to the form -7^ + hcx'^w = 0. ax Hence investigate the conditions of integrability. The symbolical form is w + jTrf. — tt e^'^+'^i^ w = ; and this may either be reduced directly by Prop. iii. to a form integrable by Prop, i, or, by assuming (w + 2)^=2^', converted into a particular case of Art. 7 in the Chapter. 458 EXERCISES. [CH. XVII. u 11 (1 uU 4. The equation -^ + — -^ 4- &w = is integrable in finite terms if a is an even number. d u 2y uu 5. The equation ^ H ~ = hx'^u is integrable in finite 4 ( 2 "I" t) terms if m = — J". ~ , where i is a positive whole number or 0. 6. The more general equation d\i 1^ du f -, ^ c\ ax adx \ x J which includes the above, is integrable in finite terms if 2V((l-rr + 4e} I being a positive whole number or 0. (Malmsten, Cambridge Mathematical Journal, 2nd Series, Vol. v. p. 180.) Verify this. 7. As an illustration of the theory of disappearing factors, integrate the equation (^' + ^^') ^!+ {(« + S) qx' + (6 - ^ -f 1)^} ^ + [{a-\-l)qx— hi] u=0. 8. The equation (1 — ax^) -~^ — hx-^ — cy^O is inte- grable in finite terms in the following three cases ; viz. 7 1st, If - is an odd integer ; a ° 2ndly, li ./ \il \ -\ — '\ is an odd integer; is an even integer. CH. XVII.] EXERCISES. 459 9. Integrate the partial differential equation d^z d^z _2 d2 _ dj? dy^ xdx 10. The partial differential equation is integrable in finite terms if Z> = ^. -. . (Legendre. See Lacroix, Tom. n. p. 618.) Yerify this. 11. Shew that the sum of the series 1 .2...nx+2.S...(n + l)x\..+j){p + l) ... (p + n-l)x'' may be expressed in the form dYx^'-o)'''-'' x{^] ^dxj 1 — X 12. Sum the series , rx 2V 3V . ^ ^ + X + lT2 + i:273+^^- 13. The equation {a + ^^) ;i^ + (/+ gx) -^ + ngu = is integrable in finite terms if n is an integer. Apply the method of Art. 13 to reduce the symbolical equation to a bino- mial form. Or assume a + hx=t. 14. The differential equation d^u dx I , ^^du f ^„ dQ , ^ m (m + 1)) ^ can be integrated in finite terms, whatever function of x is represented by Q. (Curtis, Cambridge Mathematical Journal, Vol. IX. p. 280.) 4G0 EXERCISES. [CH. XVIT. The equation may be expressed in the form ( ^ ^\ ^ o m (tti + 1) or rS^^^l^-^.m^u^Y^'^^^^ or (^)V«'^-^.± jc^^^(^ + ^)j e/^'^-t^^O. Let e''*S'^*M=v; then compare the resulting form with Ex. 8 of the Chapter. 15. Shew generally that, if we can integrate the equation we can integrate f {-^ -{■ Q\u -\- <^ {x)u = X. 16. We meet the equation ^ J -2- _ ?/ = in the theory of the elliptic functions (Legendre's modular equation). Shew that it is not integrable in finite terms, but is integrable in the form y = A + B log c, where A and B are series expressed in ascending even powers of c. 17. Prove the following generalization of Prop. ill. 18. Prove the following still more general theorem, F{D, 4, (D) e'») = P,|iJJ F[D, ^ {B) e"^] P, tgj . ( 461 ) CHAPTER XVIII. SOLUTION OF LINEAR DIFFERENTIAL EQUATIONS BY DEFINITE INTEGRALS. 1. The solution of linear differential equations by definite integrals was first made a direct object of inquiry by Euler. His method consisted in assuming the form of the definite integral, and then, from its properties, determining the class of equations whose solution it is fitted to express. Laplace first devised a method of ascending from the differential equation to the definite integral. And Laplace's is still the most general method of procedure known. Its application is however not wholly free from difficulties, due partly to the present imperfection of the theory of definite integrals, partly to an occasional failure of correspondence in the conditions upon which continuity of form in the differential equation and continuity of form in its solution depend. Indeed it ought never to be employed without some means of testing the result a posteriori, e.g. by comparison with the solution of the proposed differential equation in series. Frequently indeed it is possible to deduce the solution in definite inte- grals from the solution in series without employing Laplace's method at all. Laplace's method is applied with peculiar advantage to equations in the coefficients of which x enters only in the first degree, and of which the second member is 0. Expressing any such equation in the form '^^(3»+^(S"=<^ «' we must assume u=^le'^'Tdt, T being a function of t, the form of which, together with the limits of integration, must be determined by substituting the 462 SOLTJTION OF LINEAR DIFFEEENTIAL [CH. XVII I. expression for u in the proposed differential equation. Effect- ing this substitution, we have a result which may be thus expressed, or, since Le^*^(0 Tdt+[e'''^lr(t)Tdt = (2). Of this however, the first term is, by integration by parts, reducible to the form €^'' Thus, (2) assumes the form ^4> it) T-je" {I [

(t) T = 0, The former of these equations has reference only to the limits ; the latter, expressed in the form gives on integration, and determines T in the form ART. 1.] EQUATIONS BY DEFINITE INTEGRALS. 463 Thus, we have «=^J^I^'^' • W' the limits of integration being determined by the equation /f+JH¥t=0 (5)^ Should this equation have 7i distinct roots, these may evidently be so disposed as to give n — 1 distinct particular integrals. Such is the general statement of Laplace's method. Applied to an equation in the coefficients of which the highest power of X involved is the n^^, it would make the determination of T depend on the solution of a differential equation of the w*^ order. Other practical limitations may be noted. For in- stance, the method is only directly applicable to the expression of integrals which produce on development series of a certain form. Thus, if we develope the exponential in the assumed expression for u, we have u = JTdt + X JTtdt + ^ JTfdt + &c., an expansion in which positive and integral powers of a; alone present themselves. Integrals of different forms may, however, by preparation of the differential equation, be brought under the dominion of the method. These and other points we pro- pose to illustrate by the detailed examination of a special but very important example, particular forms of which are of very frequent occurrence in physical inquiries. We shall first, in accordance with what has above been said, determine the different kinds of solution in series of which the equation admits. This part of the investigation is intended to be supplementary to Art. 9 of the last Chapter. . x. (iiven w -^^ -\- a-j q xu = i), ax" ax -^ 464) SOLUTIONS EXPRESSED BY SERIES. [CH. XVIII. Solutions expressed hy Series. 2. The symbolical form of the above equation is "- i>(D + «-l) -"" = " («)• Hence, if an integral be expressible in the form Sw^^c"*, the law of formation of the coefficients u^^ will be "^ m(m + a-l) ^^' while the lowest value of m will be 0, or 1 — a. Thus, except in a particular case to be noticed hereafter, the complete in- tegral will be The two series in the general value of u are evidently con- vergent for all values of x. As this question of the conver- gency of series is sometimes important in connexion with the solution of differential equations, the reader is reminded that according as, in the series of terms or groups of terms 11 the ratio — - tends, when n is indefinitely increased, to a limit less or greater than unity, the series is convergent or divergent; when the ratio is less than unity but tends to unit}/, we must apply a system of criteria developed by Professor De Morgan {Differential and Integral Calculus, p. 325*). * That this system virtually includes previous special results has been proved by Bertrand {Liouville, Tom. vii. p. 35) ; that it is a legitimate deve- lopment of the fundamental principles of Cauchy has been established by Paucker [Crelle, Band xlii. p. 138). ART. 2.] SOLUTIONS EXPRESSED BY SERIES. 465 When a is an odd integer, tlie general integral will involve a logarithm. In particular if a = 1, we shall have u = a^-\- a^x^ + a,a;' + &c. + log x(h^ + h^x"" + l^x'' + &c.) ... (9), (Xq and h^ being arbitrary constants, and the succeeding coeffi- cients determined by wV + 2m&^ - 2X1-2 = 0, m'6^ - q\r,_^ = (10). The symbolical equation (6) indicates by its form that there are no solutions expressible in descending powers of x, and infinite in one direction only — i.e. beginning with some finite exponent, and presenting a series of exponents thence descending. But the equation may be traDsformed so as to admit of a solution of this kind. For, assuming u = e~'^''v, we shall have ^M^^^~' ^^^ dx~^^'^^ ' and of this the symbolical form will be found to be X>(i) + a-l)v-22(X> + |-l)e^^ = (11); whence, if v be developed in a series of the form 2^^m^™ the law of derivation of the coefficients will be m(m + a - 1)?;^ - 22'(m + ^ - 1) ^m_i = ^• It follows from this that there will be two ascending and convergent series for v, and one descending and divergent series. The law of the latter series is, by changing m into m 4- 1, more conveniently expressed in the form, [m + 1) (w + a) V = V 2q(m + ^j B. D. E. 80 466 SOLUTIONS EXPRESSED BY SERIES, [CH. XVIII.' a Hence, the first exponent will be — ^, and the ultimate value of u will be If we assume u—^'^v^ and proceed as above, we shall obtain for V the symbolical equation, i)(i) + «-l)v + 2^(Z) + ^-l)e^.'?; = (18), A and as this differs from the previous equatie-n for v, only by a change of sign affecting ^, we at once deduce a second value of u^ in the form a fa ^\ a fa ^\fa ^\fa ^ r. «r ^ 2"^V 2 2+^) 2-^2"^] 1 [ 1 .2qx 1.2. 4)«-V^ (17), •J and this, as its form suggests, and as we shall hereafter shew, is an expression for the particular integral represented by the first convergent series in the general valiie of u, given in (8). To deduce another integral, let us in the symbolical equa- tion (6) assume u = e^'^'^^^v. We find '- inj-a)i> ^''^-' (18). Hence, a value of v may be determined from that of u by changing a — 1 into 1 — a; i.e. by changing a into 2— a. Thus we have, for the second particular integral, u = C^x'~'' I e'^^ c°« ^ (sin ^) 1 - « d0, *' provided that 2 —a he positive. Hence, if a lie between and 2, we have for the complete integral. u = cJV'^^^^ (sin ey-'d9 + C^w'-'^r e'i'''^^'^ {siB.ey-''de...{19). If a = 1, the two particular integrals in the above expression merge into one. To deduce the true form of the general integral, we may proceed thus, u= r6'?^cos^{(7,(sin^)«-'+ C^ {xdndy-'']de, ^ {"" .on^AAt' /iNa-1 T,(sin^)"-'-(.Tsin^y-"l ,. -\ e3^^os^j^(sm^) ^-^-B- — ~ — ydO, on replacing C^ and C^ by two new arbitrary constants, xi and B. ART. 4.] BY DEFINITE INTEGSALS. 469 Now when a = 1, we find by tlie usual mode of treating vanishing fractions, (sin(9)«-^-(^sin6>y-'* . , . . ^.,, ^_1 -=log{^(sm(9)^}. Thus, u = [e^^^os^[^ + ^log{^(sin^)')]cZ^ (20). ♦'0 This is the complete integral of the equation ^S?+^-2^" = <' (21)' and a similar form exists for all cases in which a is an odd integer. 4. We proceed to the cases in which a is fractional and does not lie between the limits and 2. By the application of Props. II. and III. Chap, xvii., this case can be reduced to the case in which a does lie between the limits and 2. First, suppose a negative; then we may assume a=d — 2n, where a lies between and 2, and w is a positive integer. In this case, the first term of (19) will need transformation. Now the symbolical equation (6) becomes D{D + a - 2n - 1) Hence, if we assume ,2 y=0. ^ 20., _ I){B + a-l) we shall have =(4+"'-0(^i+«'-«)-(^i+'''-2™+0"-(22)' in which v=cJ e'^^cose(sin^)«'-iJ^ (28). Jo 470 SOLUTION BY DEFINITE INTEGRALS. [CH. XVIII. And this particular expression for u must replace the first term in the general value of lo given in (19). The differen- tiations may obviously be performed under the integral sign. As a particular illustration suppose a to lie between and — 2 J then n = l, a = a — 2, Avhence d , ^ J , , ^ -7- + a — 1 = ^^- + a + 1. X ax ax The particular value of u which must replace the first term in the general value (19) will therefore be u = C, rfx^ + a + l^e^'^'^'^ismOy'de. Effecting the differentiations, and substituting in (19), we have, for the general value of u, u=C, [ V^o^^ {qx cosd + a + 1) (sin e^'dO Jo + c^x'-'' [ V '^^^ ^ (sin ey-^de. Secondly, when a is greater than 2, the assumption u = e^^'^'^v, i. e. u = x^'^'v, in effect converts a into 2 — a. Compare (6) and (18). In effect, therefore, it converts a^ into a negative quantity, and reduces the present case to the preceding one. It remains only to notice that when a is an even integer, the complete integral is expressible in finite terms. Chap. XVII. Art. 3. Collecting these results together, we see that, according as a is an even integer, a fraction, or an odd integer, the complete integral is expressible in finite terms, or by definite integrals producing on development two algebraic series, or by definite integrals producing on development two series, one of which is multiplied by the factor log x. We propose before going farther to verify these results. ART. 5.] VERIFICATION. 471 Verificcction: 5. If- in tlie solution (19); we deyelope the exponentials, and for brevity write "(008(9)'" (smey-'dd^'A^ r (cos^)"; (sin Oy-'dd = 5,,... (24), we shall have ^ = ^1^ T "f " ^"^" + O.oj^"'^ 2 -, f" 5'^a?" (25), ^ 1.2...77i^ ^ 1.2...m^ ^ ^ the summation denoted by S extending to all positive inte- gral values of m, from m = to m = oo . Thus the general value of u is expressed by two series, whose equivalence to the series given in (8) it remains to establish. Now, when m is odd, A,^ = 0, B^=0, the positive and negative elements in each integral mutually destroying each other. Again, by a known formula of reduction, (cos a) (sill 6) dS ~- — ^ ^ — - J ^ ^ ' m + n + - — (cos e)"^-' (sin er de. Tn + nJ ' Supposing the limits and tt, the term free from the sign of integration vanishes at each limit when n is positive, and we have, changing n successively into a — \ and 1 — a, ^'""m + a-l "^-2* •^--^+l_tj^-2 l-^;- Now let the coefficient of x""' in the first series- in (25) be represented by u^, then A n^ A /7"*~2 n^^C.^ f" , w..= a U.2...m' '"-^ ^1.2...(m-2)' therefore ^ = -^2^--__ = ^' by (26). M^_2 m (w - 1) ^„_2 m (m + a - 1} -^ ^ ^ 472 SOLUTION BY DEFINITE [CH. XVIII. Now this is the law of the coefficients assigned in (7). And just in the same way may the second series in (25) be verified. Thus the development of the general solution (19) produces the two convergent series of the solution in Art. 2. The verification of the solution (20), though somewhat more difficult, may be effected on the same principles. Developing the exponential, and assuming r (cos er de = E^, r (cos ey' (log sin 6) d9 = i^,, ' Jo. •'0 we shall have ^^=:sf ^if" +^-^|^)gy+iog^s^-"-gy...(27), \l.z...m 1.2...m/^ ° 1.2...m^ ^ ^ the summation extending to all even integral values of m, from m = to m = 00 . Now it may be shewn that and it will be found that these relations establish, for the coefficients of the series involved in (27), the same laws of successive derivation as are assigned in (10). The verification of the solution (22) involves no difficulty. Solution hy Definite Integrals resumed. 6. In Art. 8, we found for the equation of the limits, e-'[f-q^f^O (29), from which, in order to determine the limits in perfect in- dependence of X, we rejected the factor e^. In the discussion of the same problem in the great work of Petzval*, now in course of publication, that factor is retained, giving, according * Integration der Unearen Differentialgleichungen mit Constanten und vcr'dnderliclien Goefficienten. [The second volume concluding the work was published in 1859.] ART. 6.] INTEGRALS RESUMED. 473 as X is positive or negative, the additional limit — oo or oo . And thus the following solutions are arrived at, viz. : i^ = cS €^{f-qY~'dt+ C,re'[f-q'f~"dt (30), when X is positive, and u = G, r e^' (f - q'f~' dt-^cTe^{e-^ qj ~" dt (31), J -q J q when X is negative. It will be observed that it is in their second terms that the above expressions for u differ from the expression given in (19), and the question arises, what do those second terms really represent ? We propose here to consider this question. Supposing X positive, we have to examine the term Cj'^if--q'f~'dt J -oo Now this expression, on assuming t = — q (1 + ^), so as to make the limits of integration and oo , -and performing re- ductions affecting only the arbitrary constant, becomes •I or, (7e-^^[ €^'^(2e + 6^f~'d$ (32). •'0 It is easy to see that this cannot produce either of the par- ticular integrals represented by ascending developments in (8). For, if we develope the exponential under the sign of inte- gration, the coefficient of of in the factor represented by the definite integral, will be 1.2...mj„ ^ ' But, m and a being positive, it is manifest that the expres- sion is infinite. 474 SOLUTIOJf BY DEFINITE [CH. XVIII. "We may, however, expand the definite integral in descend- ing powers of x. Developing the binomial in ascending powers of 6, and integrating by the well-known theorem r(x) {S2) assumes the form Ce-'' T^ + ^ ^ ^^/ ^ + &c. Now observing that r(^ + lj=^rf^J &c., substituting and merging the common factors in the arbitrary constant we have which agrees with (12). Exactly in the same way Petzval's second integral for the case in which x is negative, represents the other descending and divergent series (54). 7. We thus see the true nature of the distinction between Petzval's form of solution and those obtained in Art. 2. The latter represent the two converging and ascending series derived immediately from the differential equation. The former represents one of those series accompanied by the divergent series derived from a transformed differential equation*. * Spitzer, in a recent Memoir in Crelle's Journal (Vol. liv. p. 280), shews that when the coef6.cients of the differential equation satisfy the condition a-^h.^ - a^^ = h^, the solution wlLI be where U^ = &2U2 + \u + 5o , log ( YU^) = J«2^ «i^i±^o ^„^ ART. 8.] INTEGRALS RESUMED. 475 It is known that in the employment of divergent series an important distinction exists between the cases in which the terms of the series are ultimately all positive, and alter- nately positive and negative. In the latter case we are, according to a known law, permitted to employ that portion of the series which is convergent for the calculation of its entire value. Now, a being positive, the series (12) assumes this character when x is positive, the series (14) when x is negative. But these are precisely the cases in which these series are represented by Petzval's integrals. When, for the calculation of an element dependent on the solution of a differential equation, ascending and descending series are both employed (the former for small, the latter for large values of the independent variable), it is necessary to determine the connexion of the constants. For this purpose the expressioDS of the series by definite integrals may be of importance. On this, and on other points connected with this subject, the reader is referred to two most instructive Memoirs by Prof. Stokes* in which some of the equations of this chap- ter are applied to physical problems. Partial Differential Equations. 8. Some of the most interesting applications of the above method occur in the solution of partial differential equations. The following is an example. Ex. Kequired the most general solution of the equation d\ d\ dhi _ ^ d^ dy^ dz^ ' and tlie limits are giyen by The deduction of this as a limiting case of the general solution may serve as an exercise to the student. It will be proper to assume a^ + lj^x—v as the independent variable. Spitzer expresses surprise that Petzval has not arrived at the above solu- tion. We see however that it has no proper place in Petzval's actualscheme. * On the Numerical Calculation of a Class of Definite Integrals and Infi- nite Series. Cambridge Philosophical Transactions, Vol. ix. Part i. p. 166. On the Effect of the Internal Friction of Fluids on the Motion of Pendulums. Ibid. Part ii. p. 8, 476 PARTIAL DIFFERENTIAL EQUATIONS. [CH. XVIII. which can be expressed in terms of z and r, supposing This equation, with its supposed condition, presents itself in the problem of determining the attraction of a solid of revo- lution on an external point, and in the problem of the motion of an incompressible fluid, disturbed by the motion of a solid of revolution in the direction of the axis of revolution z. The transformed equation is easily found to be d^u du d^u _ , . dr^ dr dz^ ^ ^' Now the solution of the equation d?u du „ ^ is d Hence, replacing g' by y- , and A and B by arbitrary func- tions of 0, we have, for the solution of (34), u = [V" '^'^'^'^ [^ {z) + f (z) log {r (sin 0)'}] dO, or, by the symbolical form of Taylor's theorem, u^\ ^[z + r co^ 6 V(- 1)} de J -{- r ir [z + r COS ^^(-1)] log [r (sin ey]dO (35). Such is the complete integral. In all physical problems involving partial differential equa- tions the determination of the arbitrary functions so as to satisfy given initial conditions is a matter of great importance, and sometimes, where discontinuity presents itself, of great ART. 9.] PARSEVAL'S THEOREM. 477 difficulty. But though some general principles might be stated, the subject is best studied in the concrete application. In applying the above solution to the problem of attraction it is required to determine the arbitrary functions so that when r = we should have u = F {z). Now, since, when r = 0, log r is infinite, it is necessary to suppose -v/r [z) = 0. We have then F(z)=r(i>{z)de = 'ir^ (z). Thus the solution under the proposed limitation becomes - ^F[z+rcos0^J{-l)]de, IC= - ParsevaVs Theorem. 9. Equations whose symbolical form is binomial generally admit of solution by definite integrals. Pfaff's equation has thus been treated by Euler. (Lacroix, Tom. ill. p. 529.) The very beautiful theorem of Parseval, which makes the limit of the series AA' + BB'+ CO' +&c. dependent upon the limits B' (7' of the series A+Bu-\- Cv? + &c. and A' -^ H -r + &c., should be noticed. Suppose that, for all values of u. real and imaginary, A-\-Bu-\- CvJ" ... = (^(ii), a:-\-~+~^...=^{^u). u u Then, multiplying the equations together, AA' + BB'-\-CG'^-.,,-vt{a,y^+^J^ = {u)^{u). Assume, in succession, w = e^^("^^ and w = e'^^^'^^, and add the results. 478 SOLUTION OF DIFFEKENTIAL EQUATIONS [CH. XVIII. We find 2 {AA' + BB'+CC'+...) + 2t K, cos mO) + 22 (I3„, cos mO) Now multiply by d0, integrate between the limits and ir, observing that I (cos m9) dd = 0, and divide the result by •'0 2-77, then AA' + BB' + ,.. = ^j\4> 1^'^^"'^} f (e'^'"'^} + ^ {e-M-i)} ylr {e-^V(-i)}] c^6> (36), which is the theorem in question. Solution of Differential Equations hy Fourier s Theorem. 10. As Fourier's theorem affords the only general method known for the solution of partial differential equations with more than two independent variables (and such are the equa- tions upon which many of the most important problems of mathematical physics depend), we deem it proper to explain at least the principle of this application, referring the reader for a fuller account of it to two memoirs by Cauchy*. As a particular example, let us consider the equation d\i i^(d\i dhi d\\ ^ .^^ d?-^W+df+w)^^ ("')• Let u== (j> {x, y, z, t) represent any solution of this equa- tion. By a well-known form of Fourier's theorem, ^ (x) = i- r r dadXer'^^^f^-'^ ci> (a), ■^'* J -00*' -00 * Sur V Integration d'Equations Lineaires, Exercices d^ Analyse et de Physique Mathematique, Tom. i. p. 53. Sur la Transformation et la Eeduction des Integrales Genirales d''un SyS' teme d^Equations Lineaires aux differences 'partielles. Ibid. p. 178. AET. 10.] BY foueier's theoeem. 479 successive applications of which enable us to give to u the form 00 li = — Ifjllf^^^^''^ ^ i^^ ^y ^' ^) dadhdcdXdfich (38), — 00 where A= (a —x)\ + {h — y) fjL+ (c — z)v. Substituting this expression in (37), and observing that from the form given to A we have ^ . d^ . d' di we have (£+|+S^^^'""=^^^'-'(-^^-'^'-^^)' 00 — oo <^ being put for ^ (a, h, c, t). This equation will be satisfied if ^ be determined so as to satisfj the equation Hence, integrating and introducing arbitrary functions of a, h, c in the place of arbitrary constants, we have the par- ticular integrals, = 6-W(-) ^^ (a, 5, c), ci> = €--V(-) ^^ (a, l,c),.. (39), where5=(V + /^' + 0i Substituting the first of these values in (38), and merging the factor — -^ in the arbitrary function, we have OTT 00 ^^111 I 11^^^^'''^^^^''^'^' ^^' ^' ^^ (^acZ66?e£ZX6Z/^c^z^ (40), — 00 a particular integral of the proposed equation. It may easily be shewn that the employment of the second value of ^ given in (39) would only lead to an equivalent result. 480 SOLUTION BY FOUEIER'S THEOREM. [CH. XVIII. To complete the solution, we observe that if, representing d? d^ d^ -^ + -yii + 3^2 t>y 11, we make t = e^, so as to reduce the given equation to the symbolical form, TT then, by Propositions ii. and III. Chap. XVIL, the transforma- _adv dv .,, . tion ^^ = e 7^ = -t; , will give du dt which is of the same form as the equation for u. Hence, V admitting of expression in the form (40), we have, on merely changing the arbitrary function, u = j^ [fffff e(-^^^'^*W(-^) f^ {a, h, c) dadhdodXdiJLdv ... (41). — GO The complete integral is thus expressed by the sum of the particular integrals (40) and (41). The sextuple integral by which the above particular values of u are expressed admits of reduction to a double integral leading to a form of solution originally obtained by Poisson. Cauchy effects this reduction by a trigonometrical transformation. It may be accomplished, and perhaps better, by other means ; but this is a matter of detail which does not concern the principle of the solution. We may add, that when the function to be integrated becomes infinite v/ithin the limits, Cauchy's method of residues should be employed. The reduced integral in its trigonometrical form, tofjether with Poisson's method of solution, which is entirely special, will be found in Gregory's Examples, p. 504. Cauchy's method is directly applicable to equations with second members, and to systems of equations. The' above example belongs to the general form d\b rr dt' ' ART. 10.] MISCELLANEOUS EXERCISES. 481 d d d where jff is a function of j- , -r , -r • For all such equations Cijc ay ctz the method furnishes directly a solution expressed by sextuple integrals, which are reducible to double integrals if H is homogeneous and of the second degree. In the above example the double integration proves to be, in effect, an integration extended over the surface of a sphere whose radius increases uniformly with the time. Integrals of this class are pecu- liarly appropriate for the expression of those physical effects which are propagated through an elastic medium, and leave no trace behind. MISCELLANEOUS EXERCISES. 1. The complete integral of the equation dhi is expressible in the form u = Ae^"" + Be ^'', A and B being series which are finite when n is an integer. (Tortolini, Vol. V. p. 161.) 2. The definite integral I co^[n[6 — x^m 6)]d9, can be evaluated when w = + ( i + - j, where ^ is a positive integer or 0. (Liouville, Journal, Tom. vi. p. 36.) Representing the definite integral by u, it will be found that u satisfies an equation of the form y^ =[A + —^\u. The subject of the evaluation of definite integrals by the solution of dif- ferential equations has been treated with great generality by Mr Russell {Philosophical Transactions for 1855). 3. If t; = a be the equation of a system of curves, v being a function of oo and y which satisfies the equation -7-^ + -j-^ = 0, and if w = /3 be the equation of the orthogonal trajectories of the system, then u may be found by the integration of an B.D.E. 31 482 MISCELLANEOUS EXERCISES. [CH. XVIll. exact differential equation of the first order, and when found will satisfy the equation -^^ + -j-r^ = 0. The above theorem is applied by Professor Thomson to the problem of determining the forms of the rings and brushes in the spectra produced by biaxal crystals. {Camhridge Journal, 2nd Series, Vol. i. p. 124.) 4 The normal at a point P of a plane curve meets the axis in G, and the locus of the middle point of P(r is the parabola y^ = Ix. Find the equation to the curve, supposing it to pass through the origin. ( Cambridge Problems.) 5. The normal at any point of a surface passes through the line represented by -j = —=-. Find the differential equation to the surface, and obtain the general integral. (76.) 6. Prove that the differential equation of the surfaces c^enerated by a straight line which passes through the axis of z, and through a given curve, and which makes a constant angle with the axis of z,\s> 7. Integrate the above equation. 8. Express by a definite integral the series, Form the differential equation by Chap. xvii. Art. 11, and then apply 2 r- Laplace's method, Chap. XVIII. The result is m - — ^ cos{xcos6)dd. (Stokes, Cambridge Transactions, Vol. ix. p. 182.) 9. Hence express the series in a form suitable for calcu- lation when X is large. Proceeding according to the directions of Chap, xviii. the complete inte- gral of the differential equation expressed by descending series will be u = x~l{[A co^x-\-B sina;)i?+(^ Binx- B cosa;)/S}, 1^ 33 ]^2 39 52 72 Where J?^l - ^-^^ + Yr^T^AJ^^^-^""- _^__1^3^5^ 1.8x 1.2.3{8a:)3^ CH. xviil] miscellaneous exercises. 488 The values of A and B for the particular integral in question will be A =B = Tr~i. These are deduced from the consideration that, when x tends to infinity, we have, in the limit, 2 rl — / 2 cos(a:cos^)c?^ = (7ric)~i(cosa; + sina;). (Ibid.) The above series occurs in several physical problems. 10. The complete integral of the equation, "" d3 + (^ + ^^) ^ -^ (/+ 5^-^ + ^^^') 3/ = 0, may be expressed by a finite formula involving general differ- entiation. (Attributed to Liouville.) aX+- — Assume y = 2;e 2 . then, by a proper determination of a and /3, the equa- tion may be reduced to the form The symbolical equation obtained by assuming a; = e^ will be binomial, and the integration in the required form may be effected by Prop. iii. Chap. xvii. 11. Equations of the form x^ g + (J, + B.x") ^ J + (^. + B,a>- + C„*-) u = 0, may be reduced to the form, **S)^+^S)^=« W' considered in Chap, xvill. Assume x^ = t, y — t^z; the determination of k will be found to depend on the equation Jc{k-l)m^ + k{m(m-l) + mAj} +Aq = 0. Petzval, Linearen Diferentialgleichungen, Pt. 1st, p. 105. Eiccati's equa- tion is included in the above. 12. Equations of the form are reducible to the form (m). (lb. p. 112.) 31—2 484 MISCELLANEOUS EXERCISES. [CH. XVIII. 13. The complete integral of the equation is 3/ = rdte'^^i {Ce''' + a^pe^^^ . . . + C^p'^eP"^^), J where p is a primitive root of p*'"^^=l, and C, C^, C^... C^, satisfy the condition G + C^ + C^ ... + G^^ = 0, but are other- wise arbitrary. (Jacobi, Crelles Journal, Vol. X. p. 279.) 14. The determination of the orthogonal trajectory of any system of straight lines on a plane, involving in their general equation one variable parameter, can be effected by the solution of an exact differentia] equation between w and y. This interesting proposition, together with the following demonstration, was communicated to the author by Professor Donkin, with whose permis- sion it is published. The equation of the given system can always be expressed in the form xsind-y Gosd=(p{d), or, putting cos ^ = w, sin^ = v, vx-uy- F [u, v) = (1), w2 + t;2_i = o (2). The equation of the trajectory will then be udx + vdy =0 (3), u and V being determined from (1) and (2) as functions of x and y. Now, if we represent the first members of (1) and (2) by F and # respec- tively, then, in order that (3) may be an exact differential equation, we must have, in virtue of (37) Chap. XIV. dFd^_dFd^ dFd^_dFd^_ dx du du dx dy dv dv dy and this will be found to be identically satisfied. Hence (3) is an exact dif- ferential equation, as was to be shewn. The proposition applies generally to the problem of involutes. Thus, the tangents to a circle being repre- sented by vx-uy = a, u'^-'rv'^—l, the equation (3) will become \x s^/ {x"^ + y^ - aP-) -ay] dx+{y\/{x^ +y^- aP') + ax]dy _^ x^ + y^ This is exact, and determines, on integration, the system of possible invo- lutes. CH. XVIII.] MISCELLANEOUS EXEECISES. 485 15. To determine the connexion of the integrals of any system of simultaneous differential equations expressible in the form dx_dF di_dF ^ dt du ' dt dv I /-, X !► (1), du dF dv _ dF dt dx ' dt dy where ^ is a given function of x, y, u and v. The complete solution will evidently consist of four equations determining X, y, u, V SiS functions of t, and four arbitrary constants. Suppose that there exists an integral of the form $ = c, where €> is a func- tion of X, y, u, V, not involving t. Then, differentiating, we have d^ dx d$ dy d^ du d^ dv ^ _i _i j_ .^ I -— Q ^ dx dt dy dt du dt dv dt ' dx dv or substituting for — , -y- , &c. the values given in (1), d^dF d^dF_d^dF_d^dF_ dx du, dy dv du dx dv dy Now this equation is identically satisfied if ^ = F. Hence one integral will be F—a, where a is an arbitrary constant. Suppose now that another integral not involving t can be found. Then representing it by $ = 6, and observing that (2) is identical with the equation (4) in the last problem, it is seen that if, from the two equations F=a, ^=b, we determine u and v as functions of x, y, a, b, the expression udx + vdy will be an exact differential. Hence, if/ {udx + vdy) —x, we have ^X ^X ,o\ ''=Tx'''-dy (^'- Now differentiating the integral F=a with respect to a, and regarding u, V, as functions of x, y, a, h, we have dF du dF dv _ du da dv da ' dF dF or, putting for — , — — their values given in (1), and for w, v their values du dv given in (3), d^X ^ , ^^X ^ = 1 or dadx dt dady dt 486 MISCELLANEOUS EXERCISES. [CH. XVIII. whence, integrating, 1='^' (*'• c being an arbitrary constant. Since tlie form of x is known, this constitutes a third integral. Lastly, differentiating F=a with respect to & and proceeding as above, we find I- ^ (^'- e being an arbitrary constant. And this is the fourth integral. The above is a simple illustration of the methods of Theoretical Dyna- mics referred to in Chap. XIV. Thus the equations for the motion of a body attracted towards fixed centres (all in one plane) are d:^x_ dR d^_ dR di^~~'dx' 'df^~~'dy' R being a function of x, y, and the co-ordinates of the fixed centres. These equations may be expressed in the form dx dy dt ' dt ' du_ dR dv dR dt dx ' dt dy' Now, if we represent the function \{u^-\-v'^)+R hj F, the above equations assume the general form (1). It was intimated in Chap. XIV. that the solution of the equations of Dynamics is finally dependent on the obtaining of the complete primitive of a non-linear partial differential equation of the first order ; and this was previously shewn to depend on the integration of an exact differential equa- tion the coefficients of which were determined by the solution of a linear partial differential equation of the first order. Now all this agrees with what has been exemplified above. For the last two integrals, (4) and (5) are derived, by mere differentiation, from x, while x is found by the integration of an exact differential equation whose coefficients, u and v, are obtained from equations which satisfy the linear partial differential equation (2). The student is especially referred to the original memoirs by Sir W. R. Hamilton {On a General Method in Dynamics. Philosophical Transactions, 1834—5), to various memoirs by Jacobi contained in his collected works or scattered through Crelle's Journal, and to the recent memoirs of Prof. Donkin [On a Class of Differential Equations including those of Dynamics. Philosophical Transactions, 1854 — 5). Liouville's Journal is rich in valuable memoirs on the subject. ( ^S7 ) ANSWERS. The following table does not contain answers to all the questions proposed in the Exercises, but to a selected number of them, thought amply sufficient for ordinary requirements. CHAPTER I. 2. (1) 2/=^^+V(l+/). (^Here,p = ^|). (2) p-ay = €"''. (3) {1 -}- w'') p + y = isin'x. (4) a;p + y = y^ log x, (5) yip^ + "Ixp = y, (6) y=xp+4>{p). 3. (1) and (2)3+^^ = 0. (3) ^^ g + (y - 43 = 0. G. (1) [x - ay + [y- hf = 1. (2) hx-ay = ah {xy - 1). „ m 2m , m f,[ 2m\ 8. x-^,^a.y--=h,x-^,^f[^--^). CHAPTER II. 1. (1) log:r2/ + aj-2/=c. (2) log^-L^ = c. (3) (H-cc^)(l+3/^) = c^^ (4) 7(TT^ - \ l°g (1 + 2/') - l°g (y + V(l +2/=)l = 0. (5) cos y = c cos a?. (6) tan x tan y=c. 2. Yes. 8. (1) 2/ = c6"^ (2)^ = ce"^®. (3) x'^c^+2cy. (4) ^=cr"*. (5) {y+xf{y+2xY = c. 4. (1) x^xy + 2f^x-y = c. (2) (y-^+l)'(i/ + ^-l)' = c. 488 ANSWERS. 5. y = Cx" + :r^ . ^ 1- a a X 6. [^) y=ax + cx^{l-x'). (3) 3^=ce V(i-.^) + ____^^. (4) 2/=sin^-l4-ce"^'"". (5) ?/ = tan~'^-l + ce"*^"'*. 10. (1) ^ = {c V(l - ^') - a}"'. (2) s' = ce""^ _ ^ + "^ 1 a c^ ' a (3) r = {ce'^'+2(2^'+l)r^. (5) y=(c:?;+log^+l)-\ CHAPTER III. 1. x' + (jxy + y'=C, 2. x'-y^ = cx. 3. x^-y^ = cy\ 4. — TT^ + tan ^ ^ = c. 5. a? + ve^ = c. 2 ^ ^ 6. e'^ (a?^ + 2/^) = c. 7. sin (wa? + m?/) + cos {mx + n^/) = c. 9. V(l + ^'+ 2/') + tan-^-=c, sin-V(^' + 2/') + sin"^- + e^ = c. v r ^y x"" 10. Assuming: ^ = v, we have r-^ = — |- (7. ^ a?" Jc- bv^ a CHAPTER jy. 4. xyf(x' + xy-y''). Complete primitive is x^-^ xy — y"^ — c. 5. (1) InteOTatinoj factor, — rr^ sr . Solution, a;^=cV 2 cv- ^ ^ o o '£C\/(« +3/) "^ (2) Inteojrating' factor, _r - „ — ^^ « . ^ ^ o 5 '2x''+Sxy + y^ Solution, {y + cc)^ (y + 2xy = c. W ^ = ^y(i + ^)- (^) ^3/cos^ = c. 6. y = cx is tlie complete primitive. 7- (1) rTTTii^- (2) ^ xy{xy+l)' x'y'-k-x'y ANSWERS. 489 CHAPTER V. 1. (1) e^ (2) i 2. y-\ (3) I and \. ju y ax X 4. (2) y-'i\ (3) y-\\ (4) [l+y'-xY- (5) (^^ + 2/r. (6) (^ + 2/ + ^3/r- (7) {x + yT. 5. e-(a^^ + ^y^ = ^ 7. If ^ + P=?/ the equation becomes -^ ■\-2Pz = — z^, which is of the general form of 6. 9. When ^^ Q = -f . ^. Then fix) = - '^. CHAPTER YI. Equations 1 to 5 must be reduced to the form x-^ — ay ■\-'by^ — cx'^'', of which the solution is according as h and c are like or unlike in sign. In 1 we find 1 = 1, and the solution by {A) is y = a-] , where y^ is given by changing, m the first of the above solutions, a into — a,b into 1, cinto 1. In 2,i=2; apply {A). In 3 apply {B). 7. ^/(fi^ — 4a7) + n(i + ^) = 0, i being any integer, posi- tive, negative, or 0. 9. x^-(2Ah + l)y+by' = ex"'"-', where ^ is a root of the equation bA^ + A - h = 0. 10. Compare with p. 95. 490 ANSWERS. CHAPTER YII. 1. (y-2x-c) {y^'^x-c) =0. 2. {y -a\ogX'-c){y-\-alogx-c) =0. 5. Eliminate p by means of a log_p + 2hp + c = ^. 2 8. By 2/ = ^ +1 V(l +/) - i log \p + V(l + /)^)) + c. 12, Complete Primitive y = cx-\-c — c^. Singular Solution y=^- — -j—^* 13. Complete Primitive, y = ex -}- a/ (h^ — d^c^). 2 2 Singular Solution, r, + -^ = 1- 14. x" + v'' = ex. 16. Eliminate p loj x = -^^^p — 2^ (c + a siiT^p) . 17. By a? = ..^-^ ,^ (c -f - + a tan"' p). 19. (a'-a)^+{2/-/(a)f=L 21. ax- yf {a) = af (a) (xy-1). CHAPTER YIII. 4. Singular Solution x = a, 6. Differential equation, ^ 2 V (cc — a) ' 4 ■ 10. (1) xy^l, (2) g)'±g)' = l. (3) 3/ = ^- 11. Particular Integral. 13. Singular solution y=0; complete primitive ?/=e'^'' ~^\ AXSWEES. • 491 — x^ 16. (1) Envelope species, y= — - . (2) Envelope species, 'if — 4. 21. ^ = a^e^ + ^ 6^^+ 6. a 2 23. z =xy-^y Jix^ - (t) + h and z = ^-^ -f _^- + &. •^ "^ a x — y CHAPTER XV. 8. z = <^[x^-ay)+ysj{-l-a'). ANSWERS. 495 CHAPTER XVI. mx 4. w = e'-(^ + |cc + ()+ce'^+c'6'^ 711 — 5?7^ + 6 t ? 8??^ sin ma? — (m^ — 2) cos mx _^ _„, 8. « = .«^(|)+.t(|). 10. w = cos(7ilogaj)^(^|,|j + sm(7ilogic)>|r(^|, 5^ 11. Assume -j- -\-X — it. ax CHAPTER XVII. 1. w = ce"" [x-V)- c'e^ {x + 1). 9 —( 'i ^ d\G + c log a; \ c^x""* dx) 1 —X ,a-6+l ,dr/ a;* (1 + ^-^j d ^-- '-i)"^- 496 NOTE. Note on Art. 4 of Chap XIV, page 327. [The language here used is not quite satisfactory. It is asserted that from equation (7) we must have the two equations (8) since ^'{v) is arbitrary. But by the same argument it would follow on page 326 that we must have du du dv dv dx dz dx dz . du du . ^ dv dv also 3- + ^-9' = 0, and -— + -^ fl' = 0. dy dz dy dz In fact, instead of saying that the two equations (8) must hold, we ought to say that we may consistently with (7) assume them both to hold. Then it will follow that the relations (9) must be consistent with the relation pdx + qdy = dz. This is sufficient to enable us to deduce the equation (10). Traces of the same inaccuracy of language will be found in other parts of the Treatise, though not so decided as in the present passage : see pages 333 and 363. This correction is due to the Bev. H. W. Watson, formerly Fellow of Trinity College.] CAMBRIDGE ! PRINTED BY C. J. CI*AT, M.A. AT THE UNIVERSITY PEESS. "^1 DATE DUE MAY 19 |< 90 ,q ]m'} lAV \ -^ 1 ! UNIVERSITY PRODUCTS, INC. #859-5503 ! BOSTON COLLEGE '^ lOTfl^^ ^1 3 9031 01549563 3 BOSTON COLLEGE LIBRARY UNIVERSITY HEIGHTS CHESTNUT HILL, MASS. 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