CORNELL UNIVERSITY LIBRARIES Mathematics Library '■"'•^ii-fi Hall CORNELL UNIVERSnv LIBRARY 3 1924 059 551 295 DATE DUE ^. ^B*^ J# CAYLORD PNINTEOINU.S.A. Cornell University Library The original of this book is in the Cornell University Library. There are no known copyright restrictions in the United States on the use of the text. http://www.archive.org/cletails/cu31924059551295 Production Note Cornell University Library produced this volume to replace the irreparably deteriorated original. It was scanned using Xerox software and equipment at 600 dots per inch resolution and compressed prior to storage using CCITT Group 4 compression. The digital data were used to create Cornell's replacement volume on paper that meets the ANSI Standard Z39. 48-1984. The production of this volume was supported in part by the Commission on Preservation and Access and the Xerox Corporation. 1990. /^crti^ 6a^Uzi>U7?UAA^ /f^r CORNELL UNIVERSITY LIBRARY MATHEMATICS OHDINARY DIFFERENTIAL EQUATIONS AN ELEMENTARY TEXT-BOOK WITH AN INTRODUCTION TO LIE'S THEORY OF THE GROUP OF ONE PARAMETER BY JAMES MORRIS PAGE PH.D., CNIVBRaiTY OF LEIPZIG : FELLOW BY COURTESY JOHNS HOPKINS UNIVERSITY ; ADJUNCT PROFESSOR OF PURB MATHEMATICS UNIVERSITY OF VIRGINIA MACMILLAN AND CO., Limitep NEW YORK : THE MACMILLAN COMPANY 1897 All rightt reterved GLASGOW : PRINTRD AT THB 0KIVEBSIT7 PRESS BY ROBERT MACLBHOSE AWD CO. PREFACE. This elementary text-book on Ordinary Differential Equations, is an attempt to present as much of the subject as is necessary for the beginner in Differential Equations, or, perhaps, for the student of Technology who will not make a specialty of pure Mathematics. On account of the elementary character of the book, only the simpler portions of the subject have been touched upon at all ; and much care has been taken to make all the developments as clear as possible — every important step being illustrated by easy examples. In one material respect, this book differs from the older text-books upon the subject in the English language : namely, in the methods employed. Ever since the discovery of the Infinitesimal Calculus, the integration of differential equations has been one of the weightiest problems that have attracted the attention of mathematicians. It is not possible to develop a method of integration for all differential equations ; but it was found possible to give theories of integration for certain classes of these equations ; for instance, for the homo- geneous or for the linear, differential equation of the first order. Also, important theories for the linear differ- ential equations of the second or higher orders, have vi ORDINARY DIFFERENTIAL EQUATIONS. been developed. But all these special theories of in- tegration were regarded by the older mathematicians as different theories based upon separate mathematical methods. Since the year 1870, Lie has shown that it is possible to subordinate all of these older theories of integration to a general method : that is, he showed that the older methods were applicable only to such differential equations as admit of known infinitesimal transformations. In this way it became possible to derive all of the older theories from a common source : and at the same time, to develop a wider point of view for the general theory of differ- ential equations. Only a very small part of Lie's extensive and im- portant developments upon these subjects could, however, be presented in a text-book intended for beginners. The memoirs published by Lie on differential equations are to be found in the " Verhandlungen der Gesellschaft der Wissenschaften zu Christiania," 1870-74; in the Mathematische Annalen, Vol. II., 24 and 25 ; and in his Vorlesungen iiber JDifferentialgleichungen mit Be- lannten Infinitesimalen Transformationen, edited by Dr. Gr. Scheffers, Teubner, 1891. Besides these sources of information, the writer had the advantage of hearing, in 1886-87, at the same time with Dr. Scheffers, Prof. Lie's first lectures upon these subjects at the University of Leipzig. All the methods, depending upon the theory of trans- formation groups, employed in Chapters III.-V., and IX. -XII. of this book, are due exchisively to Prof. Lie. Lie has also developed elegant theories of integration for Clairaut's and Eiccati's equations, as well as for the PREFACE. vii general linear equation with constant coefficients ; but, as an exposition of these theories requires a more ex- tensive preparation than it was considered advisable to give in a purely elementary text-book, the author deter- mined to follow, in the treatment of the above-mentioned equations, the older methods — hoping to present Lie's methods for these equations, as well as some of his more far-reaching theories, in a second volume. In the preparation of this book the author has made free use of the examples in the current English text- books : and he is under special obligations to the works of Boole, Forsyth, Johnson, and Osborne. The treatment of Eiccati's equation. Chapter VII., is substantially that given by Boole. The arrangement of the matter will be found suffic- iently indicated by the table of contents ; and an index is given at the end of the book. The articles in the text printed in small type may be omitted by the reader who is going over the subject for the first time. JAMES MOREIS PAGE. Johns Hopkins University, Baltimore, U.S.A., July, 1896. CONTENTS. CHAPTER I. GENESIS OF THE ORDINARY DIFFERENTIAL EQUATION IN TWO VARIABLES. PAGE § 1. Derivation of the Differential Equation from its Complete Primitive. Order and Degree of a Differential Equation, 1 Definition of General Integral, 3 Particular Integrals, ■ - 4 § 2. Geometrical Interpretation of the Ordinary Differential Equation in Two Variables, - - - 6 Examples to Chapter I., • - - - - - 9 CHAPTER II. THE SIMULTANEOUS SYSTEM, AND THE EQUIVALENT LINEAR PARTLAL DIFFERENTIAL EQUATION. § 1. The Genesis of the Simultaneous System, ... 10 § 2. Definition of a Linear Partial Differential Equation, . 13 The Linear Partial Differential Equation and the Simul- taneous System represent fundamentally the same Problem, - 14 Geometrical titerpretation of the Simultaneous System in Three Variables, - . - 17 ORDINARY DIFFERENTIAL EQUATIONS. PAGE § 3. Integration of Ordinary Differential Equations in Two Variables in which the Variables are Separable by Inspection, 19 Integration of a Special Form of Simultaneous System in Three Variables, 21 Examples to Chapter II., 24 CHAPTER III. THE FUNDAMENTAL THEOREMS OF LIE'S THEORY OF THE GROUP OF ONE PARAMETER. § 1. Finite and Infinitesimal Transformations in the Plane. The Group of one Parameter, - - - 25 Definition of a Transformation, 26 Definition of a Finite Continuous Group, 27 Derivation of the Infinitesimal Transformation, 29 Kinematic Illustration of a Gj in the Plane, - 32 The Increment Sfoi a Function /(arj, y{) under an Infinitesi- mal Transformation, 36 The Symbol of an Infinitesimal Transformation, 37 The Form of the Symbol when New Variables are Introduced, 38 The Development f{x„ y,) =f(x, y)+Uf. t+ U(U-/]^+ ... , and the Equations to the Finite Transformations of a G^, 40 § "2. Invarianoe of Functions, Curves, and Equations, - - 42 Condition that the Function Q{x, y) shall be Invariant under the Gi Uf, 42 The Path-Curves of a G^ in the Plane, 44 Condition that a Family of Curves shall be Invariant under a (?j in the Plane, - 47 Condition that the Equation fi = shall be Invariant under the (?i Uf, 50 Method for Finding all Equations which are Invariant under a given Gj in n Variables, 51 CONTENTS. xi PAGE § 3. The Lineal Element. The Extended Group of One Para- meter, - 54 The Infinitesimal Transformation of the Extended Gj, - 59 Examples to Chapter III., 59 CHAPTER IV. CONNECTION BETWEEN EULER'S INTEGRATING FACTOR AND LIE'S INFINITESIMAL TRANSFORMATION. § 1. Exact Differential Equations of the First Order in Two Variables, 62 Condition that a Differential Equation of the First Order shall be Exact, 63 Definition of Euler's Integrating Factor, 66 § 2. Invariant Differential Equation of the First Order may be Integrated by a Quadrature, 67 Definition of an Invariant Differential Equation, 67 To find all Differential Equations which are Invariant under a given O^, 68 The Integral Curves of an Invariant Differential Equation of the First Order constitute an Invariant Curve-Family, 73 Proof of the Theorem that every Difierential Equation of the First Order which is Invariant under a known G^ may be Integrated by a Quadrature, 73 Definition of a trivial G^, • 78 § 3. Classes of Invariant Differential Eqvalions of the First Order in Tioo Variables. The Equations Invariant under Uf=:^, The Equations Invariant under (//"= ^^y • All homogeneous Differential Equations of the First Order 79 80 are Invariant under f7/'=x=— -I- y,^, 81 ^ dy xii ORDINARY DIFFERENTIAL EQUATIONS. . PAGE The Equation (ax + by + c)dx- (a'x + b'y + c')dy = may usually be reduced to a homogeneous form, 83 The Equations Invariant under U/= -yS- + x^, - 84 The Equations fi(xy)xdy -fi(3cy)ydx = are Invariant under f7/=a;^ -y^, - - 87 ox "oy The Linear Equation y'-4,(x)y-i>(x) = (i ia Invariant under £//■= e'' . ^, - - 88 The Equation y'-^(x)y-ll'(a;)/ = may be reduced to a Linear Form, - - - 90 The Classes of Invariant Differential Equations of the First Order may be Multiplied Indefinitely, - - - - 91 To Find the (?i of which a given Differential Equation of the First Order admits, - - - 92 The Development of the General Integral of a Differential Equation of the First Order in a Series, - - - 93 Table of Classes of Invariant Differential Equations of the First Order, - - - - 96 Examples to Chapter IV., - 97 CHAPTER V. GEOMETRICAL APPLICATIONS OF THE INTEGRATING FACTOR. ORTHOGONAL TRAJECTORIES AND ISO- THERMAL SYSTEMS. Geometrical Meaning of the Integrating Factor, - 100 Application to Parallel Curves, 101 Orthogonal Trajectories, 103 Isothermal Systems, - 104 E.tamples to Chapter v., 107 CONTENTS. xiii CHAPTER VI. DIFFERENTIAL EQUATIONS OF THE FIEST ORDER, BUT NOT OF THE FIRST DEGREE. SINGULAR SOLU- TIONS. PAGE § 1. Differential Equations of a Degree Higher than the First, 109 Decomposable Equations, - 110 Equations which may be solved with respect to y, ■ 111 Equations which may be solved with respect to a;, • 112 § 2. Method for Finding the Singular Solution of an Invariant Differential Equation of the First Order, 113 Examples to Chapter VI., - - - 118 CHAPTER Vn. RICCATI'S EQUATION, AND CLAIRAUT'S EQUATION. § 1. Riccati's Equation, - 119 The Equation Xj^ - ay + 6^^ = ex" ax is Integrable when n = 2a, - - 120 This Equation is also Integrable when — = — is a Positive Integer, • ... 123 The Forms of the General Integral, 125 Application to the Equation '^ + bw' = cz"', ■ 125 § 2. Clairaut's Equation, .... 128 The Equation y = x is {U, A) = \{x,y,z)A/, ■ ■ ■ 200 Two Linear Partial Equations ^i/=0, Aif=0 which satisfy a Condition of the Form {A„A^) = \AJ+\^A,/, may always be put into such Forms that they satisfy the Condition, Mj, ^j) = 0, - - 201 xviii ORDINARY DIFFERENTIAL EQUATIONS. PAGE To Find the Common Solution which Af= 0,' Uf= Possess when (U,A) = (i, - 203 This Solution is the General Integral of a Total Equation in Three Variables, - 204 When this Solution is known, the Invariant Linear Partial Equation may be completely Integrated by a Quadrature, 205 Rfeume of the Results of the Section, - 206 § 3. Application of the Method of § 2 to the Ordinary DiflFerential Equation of the Second Order in Two Variables, which is Invariant under a known Gj, 208 Examples to Chapter XII., 211 CHAPTER I. THE GENESIS OF THE ORDINAEY DIFFERENTIAL EQUATION IN TWO VARIABLES. GEOMETRICAL INTERPRETATION. 1. In the first section of this Chapter, we .shall explain what is meant by an ordinary differential equation in two variables, and show how to derive a differential equation from its coTnplete primitive. In the second section, we shall show how ordinary differential equations in two variables may be interpreted geometrically. SECTION I. Complete Primitive. Order and Degree of an Ordinary Differential Equation. 2. An equation of the form co{x,y) = (1) is ordinarily used to express in algebraic language the fact that one of the two variables x and yisa, function of the other. If this equation also contains an arbitrary constant c, its presence is indicated by writing the equa- tion in the form co{x,y,c) = (10 2 ORDINARY DIFFERENTIAL EQUATIONS. By differentiating (1'), we obtain and the constant c may have been removed by the pro- cess of diiferentiation. If, however, (2) still contains c, it may be eliminated by means of (!') ; so that we find, either immediately after the differentiation, or after the elimination, an equation involving x, y, and -j^, of the general form 4'^'|)=<> (3) If we make use, as we shall often do, of the customary abbreviations, the last equation may be written F(x,y,y') = 0; (3) and (3) is called an ordinary differential equation of the first order in two variables. 3. If the equation (1) contains tivo independent arbi- trary constants, so that it may be written in the form w{x,y,c,d) = 0, (1") (c, d, consts.) ; two successive differentiations of (1") will give an equation containing y", from which, by means of (1") and the equation obtained from (1") by a first differenti- ation, both arbitrary constants, c, d, if they are still present, may be eliminated. We obtain thus an equation of the general form F{x,y,y',y") = 0, (4) which is called an ordinary differential equation of the second order in two variables. COMPLETE PRIMITIVE. 3 4. The equations (1') and (1") from which the diflfer- ential equations (3) and (4) are obtained, are called the complete primitives of (3) and (4), respectively. It is clear that if (1) contained three independent arbitrary constants it would give rise to a differential equation of the third order; and, in general, we see that the order of a differential equation, which is defined as that of the highest derivative in the equation, is the same as the number of independent arbitrary constants in the complete primitive. Thus, if the com- plete primitive contains n independent arbitrary con- stants, it will give rise to a differential equation of the ;^th order. The degree of a differential equation is the same as the degree of the derivative of the highest order in the equation, after the equation has been put into a rational form, and cleared of fractions. Thus the equation is of the second order, and of the second degree. From what has been said, it is seen that to find the differential equation of the n^^ order corresponding to a primitive containing n arbitrary constants, it is necessary to differentiate the primitive n times succes- sively, and eliminate, between the n+l equations thus obtained, the n arbitrary constants. The resulting equation will be the required differential equation of the n^^ order. 5. The inverse process — usually involving one or more integrations — of finding from a differential equation its complete primitive, is called solving, or integrating, the differential equation, and the arbitrary constants, which were formerly made to vanish by differentiation and elimination, now reappear as constants of integration. When the equation thus obtained contains exactly n independent arbitrary constants, it is called the general integral, or the complete primitive, of the differential 4 ORDINARY DIFFERENTIAL EQUATIONS. equation of the to* order. Thus, if F(a;,2/,3/'. ...,3/W) = (5) be a differential equation of the n*^ order, its general integral will be an equation of the form ui{x, y, Cj, ..., c„) = (6) where the c^, ..., c„ are independent arbitrary constants. It may be noted that (6) is usually referred to as the general integral of (5), when (6) is considered as having been derived from (5) ; if, however, (5) is considered as having been derived from (6), (6) is referred to as the complete primitive of (5). It is evident from the method of deriving from a complete primitive its corresponding differential equation that the general integral cannot contain more than n independent arbitrary constants ; for the general integral w^ould then, being treated as a complete primitive, give rise to a differential equation of an order higher than the n*\ 6. If a special numerical value is assigned to each of the arbitrary constants, respectively, of a known general integral of a given differential equation, the resulting equation is called a particular integral of the given differential equation. Thus the particular in- tegral is free from all arbitrary constants of integration. For example, if the general integral has the form y — mx — n = 0, then the equations y-2x-5 = 0, 2/- 3a;- 7 = 0, etc., will be particular integrals of the given differential equation. 7. We shall now apply to two simple examples the method of finding the differential equation corresponding to a given complete primitive. COMPLETE PRIMITIVE. 5 Example 1. It is required to find the diflPerential equation of the first order corresponding to the complete primitive y-cx=Q, (7) where c is an arbitrary constant. By dififerentiation, we obtain, dy- -cd:c-- = 0, or c= dy 'dx Hence, from the first equation, dy_ dx' X .(8) This is the differential equation required. If we consider (8) as given, and (7) as having been derived from it — by methods to be explained later — (7) is called the general integral ai (8). By assigning to c in (7) different numerical values, different particular integrals are obtained. Example 2. It is required to find the differential equation of the second order corresponding to the complete primitive, a^ + 2ax+y^+2b^ = 0. (a, b, consts.) By two successive differentiations, we obtain the equations x+a+yy' + by'=0, l+y'^+yy"+by"=0. If a and b are eliminated from these three equations, we find, as the differential equation required, {x' +y^)y" - 2xy'3 + 2yy'^ - 2xif + 2y = 0. 8. It has been shown that to pass from a complete primitive to the corresponding differential equation involves merely the processes of differentiation and elimination ; but since the steps of an elimination cannot be retraced, it is a matter of much greater difficulty — if possible at all — to pass from the differen- tial equation to the corresponding complete primitive, or general integral. It will be our object to show how, in a number of the simplest and most important cases, we may, from a given differential equation, deduce its general integral. 6 ORDINARY DIFFERENTIAL EQUATIONS. SECTION II. Oeometrical Interpretation of Ordinary Differential Equations in Two Variables. 9. If the ordinary differential equation of the first order in x and y, F(x,y,y') = 0, (1) be written in the solved form, ^ Xix,yy ^^^ where X and Y are supposed to be one- valued functions, it is clear that to any pair of values ascribed to x and y, a fixed value of y' will correspond. If we consider x and y to be the rectangular coordinates of a point in the plane, y' will represent the numerical value of the tangent of the angle made with the aj-axis by the straight line connecting the point {x, y) with the origin of coordinates. Now suppose the point (x, y) to move a short distance in the direction given by y' ; in the new position of the point, y' will generally have a new value. Suppose the point to move a short distance in the direction now given by y' ; in this third position of (x, y) there will be in general a third value ascribed to y': the point (x, y) can now be supposed to move a short distance in this last direction — and so on. By this means a figure will be traced of which the limit will be a curve of some kind, when the distances through which the point {x, y) is moved are indefinitely diminished. At every point on this curve the equation 2/' = J (2) is satisfied ; that is, if <^(x,y) = (3) be the equation to this curve, the equation (o = must GEOMETRICAL INTERPRETATION. be a particular integral of equation (2), or of the equi- valent equation (1). The curve traced by a point moving under the above restrictions is therefore called an integral curve of the ordinary differential equation (1). If we start from any point not on the curve (3), it is evident that by proceeding as before we get a new integral curve. We might, for instance, take as successive starting points the points on the a;-axis — provided that the a;-axis does not happen to be itself an integral curve — and it is evident that, in all, oo^ different integral curves would be obtained, one passing through every point of ordinary position in the plane. These curves must be represented by an equation of the general form, u,{x, y, c) = 0, (4) where c is an arbitrary constant, or parameter, which assumes different numerical values according as (4) is made to represent the different individual curves of the whole system of integral curves belonging to equation (1). In other words, (4) is the general integral of (1). Example. The differential equation of the first order xdy-ydx=Q, or, O X represents a system of ooi straight lines through the origin. For " is the numerical value of the tangent of the angle between the j;-axis and the line joining the point {x, y) with the origin ; and 8 ORDINARY DIFFERENTIAL EQUATIONS. as y gives the direction in which the point (x, y) is to be moved, equation (5) asserts that the point (x, y) always moves on the straight line connecting that point with the origin. Since there- fore each point of the plane moves on one line of a system of straight lines through the origin, equation (5) represents the family of 00' straight lines 1=0 (6) c being the arbitrary parameter. Thus (6) is the general integral of (5) : and the particular integrals are obtained by assigning to c different numerical values. 10. Since the complete primitive, or the general integral, of a differential equation of the second order must contain two independent arbitrary constants, or parameters, it is clear that this general integral, or, as we may say, the differential equation of the second order itself, represents geometrically a doubly infinite system of curves in the plane. Similarly, a differential equation of the third order represents a triply infinite system of curves, etc. Example. The ordinary differential equation of the second order y'=o (7) asserts that the curvature of the path along which the point {x, y) is to be moved is everywhere zero. Hence the point {x, y) must always describe a straight line, that is, the doubly infinite system of curves which satisfy the above differential equation must be the 00 2 straight lines of the plane y — mx-n=0 (8) It may at once be verified that (8) is the general integral of (7). EXAMPLES. Form the differential equations of which the following are the complete primitives, a, b, c being arbitrary constants. (1) y=cx. (2) y = cx + ^\+' circles having their radii equal to r : (16) Form the differential equation of all circles having their radii equal to r. (17) Find the differential equation of the family of straight lines which touch the circle ;r2+y2=l; and show that the circle itself also satisfies the differential equation. The equation to the tangents is fla+6y-l=0 where the constants a and h must satisfy the condition a2+62=l. (18) Find the differential equation of all the conic sections whose axes coincide with the coordinate axes : (19) Find the differential equation of all logarithmic spirals around the origin : CHAPTER 11. SIMULTANEOUS SYSTEMS OF ORDINAKY DIFFER- ENTIAL EQUATIONS, AND THE EQUIVALENT LINEAR PARTIAL DIFFERENTIAL EQUATIONS. II. We shall reserve for a later chapter the con- sideration of the genesis of an ordinary differential equation in three or more variables, when that equation is obtained from a single primitive by methods similar to those of Chapter I. It will be necessary, however, to give in Sees. I. and II. of this chapter a few propositions relating to simultaneous systems of ordinary differential equations, and the equivalent linear partial differential equations, in order to develop in the next chapter as much of the Theory of Transformation Groups as we shall need. The third section of this chapter is intended as a supplement to this chapter and to the preceding one. We there indicate, for convenience of reference in Chapter III., the method of integrating the simplest form of an ordinary differential equation in two variables, a problem which really belongs to the Integral Calculus ; and we also make a remark upon the integration of the simplest form of a simultaneous system in three variables. A theory of integration for the general simultaneous system will not be given until Chapter XII. SIMULTANEOUS SYSTEMS. 11 SECTION I. The Simultaneous System of Ordinary Differential Equations. 12. Suppose two equations of the form U{x,y,z) = a, V(x,y,z) = b (1) are given, where U and V are independent functions of x,y,z, and a and b are arbitrary constants. By differ- entiating (1) we find •(2) ^^+^dy + -dz=0, ^dx + — dy + -dz = 0, as resulting equations. But from the equations (2) we find that relations of the form <^ §y dz dUdV_d£dV dUdV_dUdV~dUdV_dUdV-^'^^ dy dz dz dy dz dx dx dz dx dy dy dx must hold : and, if we denote the denominators of these ratios, which are known functions of x, y, z, by X(x, y, z), Y(x, y, z) and Z{x, y, z), respectively, the equations (3) may be written dx _dy _dz . . T~T~"J ^^' Thus the system of equations (1), treated as simultaneous complete primitives, gives rise to the so-called simul- taneous system, of ordinary differential equations of the first order, (4). 13. This result in three variables is entirely analogous to that of Art. 2 in two variables. The differential 12 ORDINARY DIFFERENTIAL EQUATIONS. equation derived in that article from .one primitive of the form U{x, y) = a may, of course, be written in a form symmetrical with (4), dx _ dy 14. It is obvious that the results of Art. 12 may be extended to n variables. If t/j(a3j, (Cg, ... , x„)^(X]^, U^(X-^^ a;„) = (i2, ..., Un-iixi, ...,a;„)=a;„.i (5) be a system of n— 1 equations in the n variables X\, ..., Xn, the Ui, ..., Un-i being independent functions of those variables, and the ai, ..., a„_i being arbitrary constants, the system of equations (5), being treated as simultaneous complete primitives, will evidently give rise to a so-called simultaneous system of ordinary differential equations of the first order, which may be written in the form Here the X^, ..., X„ are known functions of cCj, ...,x„. In the next section we shall see how the simul- taneous system in three variables may be interpreted geometrically. SECTION II. Simultaneous Systems avd the Equivalent Linear Partial Differential Equations. 15. Equations are of frequent occurrence by means of which a relation between the several partial deriva- tives of a function of two or more variables is expressed. If / be any function of x, y, z, the general form of such SIMULTANEOUS SYSTEMS. 13 an equation, involving only partial derivatives of / of the first order, and the variables x, y, z, will be and if / be known, the values of its partial derivatives substituted in this equation must satisfy the equation identically. An equation which expresses a relation between the partial derivatives of a function of two or more inde- pendent variables — and which may also contain the independent variables themselves explicitly — is called a partial differential equation ; and the function /, whose partial derivatives satisfy the equation identically, is called the solution of the equation. The order and degree of a partial differential equation are determined just as are the order and degree of an ordinary differential equation. A partial differential equation of the first order and degree is said to be linear of the first order ; the term linear having reference only to the manner in which the partial derivatives of the solution / enter the equation. Thus the general form of a linear partial differential equation of the first order in n variables is where the X^, ..., X„ are certain known functions of the independent variables a;,, ..., a;„. We shall hereafter limit ourselves to the consideration of such partial differential equations as are linear and of the first order; since this class of equations is, as we shall see, intimately connected with ordinary differential equations. 16. The ordinary differential equation of the first order in two variables may be written in the solved form, dx _ dy _ .^. X{x,y)-Y{x,yy ^'^ 14 ORDINARY DIFFERENTIAL EQUATIONS. and an intimate relationship may be shown to exist between (1) and the linear partial differential equation in two variables, X(a;,2/)|+F(x,2/)^=0 (2) For, if a)(a;,2/) = const, be the integral of (1), we find by differentiation, ^^^^•^y-o <^) Now eliminating -^ between (3) and (1), we find as a necessary consequence of these equations the identity, ox dy That is to say, if the equation w{x, y)=c is an integral of the ordinary differential equation (1), w is also a solution of the linear partial differential equation (2). Conversely, it may be readily seen that if the function ft) is a solution of the linear partial equation (2), ft) = c will also be an integral of (1). Thus the equations (1) and (2) represent fundamentally the same problem, since to find an integral of (1) is the same as to find a solution of (2), and vice versa. 17. If the general integral of a given differential equation of the first order (1) has been put into the form Q (x, y) = c, (c = const.) we shall call the function Q(x, y) the integral-function of the given differential equation. It is a proposition of the Theory of Functions, which we shall here assume without proof, that an integral- function of a differential equation of the first order always exists, and that all integral-functions of a given differential equation of the first order must be functions of any one of the integral-functions; that is, that no differential equation of the first order, (1), can have two SIMULTANEOUS SYSTEMS. 15 iv dependent integral-functions. Thus if U and V be two integral-functions of a given differential equation of the first order, we must be able to express the one as a function of the other, say [/ = $(F). From this it follows that if we know any integral of a differential equation of the first order, containing an arbitrary constant, we may regard all possible integrals of that equation as known.* Also, since (1) always has an integral-function, though it cannot have two independent integral-functions, the linear partial differential equation of the first order (2) must always have one solution, although it cannot have two independent solutions. The whole number of solu- tions of (2), or of integral-functions of (1), is evidently unlimited; for if o) be a solution of (2), it is easy to see that any function of w, as ^{tio), is also a solution of (2). For, substituting $(«•)) in place of / in (2), we find for that equation du)\ dz dy/ but as the expression in parenthesis is zero on account of ft) being a solution of (2), the left-hand member of the last equation is zero, that is, $((u) is also a solution of (2). Since every solution of (2) is an integral function of (1), it also follows from this that the most general integral of the ordinary differential equation (1) has the form $((o) = const., where w is any integral-function of (1). *The fact that an ordinary differential equation always has a general integral is illustrated by the types of integrable equations. Chapter IV. , as well as by the development, Art. 72, of the general integral in a 16 ORDINARY DIFFERENTIAL EQUATIONS. 18. The linear partial differential equation in three variables has the form X{x, y, ^)g+ Y{x, y, z)^+Z{x, y, z)^=0, (4) and it is easy to see that the same relation exists between (4) and a system of equations of the form dx_dy_dz T~T~Y' ^^^ that was seen to exist between (1) and (2). It is shown in the Theory of Functions that there are always two, and only two independent functions of the form U{x, y, z), V{x, y, z), which, when written equal to two arbitrary constants a, b, respectively, U(x, y, z) = a, V(x, y, z) = b, (6) will give, when these equations are differentiated as in Art. 12, values for the ratios dx, dy, dz, which satisfy the simultaneous system (5). When the equations (6) are derived from (5) — by methods to be explained later — they are called the integrals of (5). By differentiation, we find from (6) BfT, -dJJ, -dU, . —-dx+^r-dy+-:—dz = 0, 3a; Zy ^ ?iz dV, -dV. -dV, . —dx+^dy + -dz = 0; and these equations, by means of (5) may be written, ?ix dy dz ox ay dz But the last two equations show that the functions U SIMULTANEOUS SYSTEMS. 17 and V, which in accordance with Art. 17 we shall call the integral-functions of (5), must be solutions of the linear partial differential equation (4). It is thus obvious that any integral-function of (5) must be a solution of (4), and vice versa. Hence we see that (4) cannot have more than two independent solutions ; that is, that every solution of (4) must be capable of being expressed as a function of any two independent solutions of (4). The whole number of solutions is, however, unlimited ; for if U and V are solutions, it is easily seen that any function of U and F, as ^{U, V) is also a solution of (4). For, substituting $ for / in (4) there results but since U and V are solutions of (4), the expressions in the parentheses are zero; ithat is, the last equation is identically satisfied, or $( C7, V) is a solution. Since the solutions of (4) are also integral-functions of (5), the most general integral of (5) has the form ^(U, F) = const., where U and V are any two independent integral- functions. 19. The equations of the preceding article, U(x, y, z) = a, V(x, y. z) = b, (6) represent two families of oo^ surfaces in space ; these are the so-called integral surfaces of the simultaneous system (5). Also the system of equations (6) may obviously be said to represent a family of oo^ curves in space — the curves of intersection of the two families of surfaces — each particular curve being obtained by assigning a pair of special numerical values to the arbitrary constants a and b. One of these curves evidently passes through every general point in space ; and at every general point P, on one of these curves, the equations (5) must be satisfied. That is to say, the tangent at the point P 18 ORDINARY DIFFERENTIAL EQUATIONS. to the curve passing through that point must have a direction of which the direction cosines are proportional to X, T, Z respectively. These oo^ curves, at every point of which the equations (5) are satisfied, are sometimes designated as the char- acteristics of the linear partial differential equation (4), which is equivalent to the simultaneous system (5). Example. As a simple example, we may suppose the equations 6) to have the forms x+y+z=a, x'^+y^+z^ = h\ (6') (a, 6^ consts.) the first equation representing a system of parallel planes, the second a system of concentric spheres around the origin. Thus the simultaneous equations (6') represent the oo^ circles cut from the 00 1 concentric spheres by the oo' parallel planes. By the method of Art. 12 we find the simultaneous system to which (6') give rise by differentiation in the form, dx+dy + dz=0 xdx+ydy + zdz=0 ; . dx dy dz whence = — 2- = . 0-y x—z y—x This is of course equivalent to the linear partial differential equation (-y)g+(--)|+(y-)|=0; and it may be readily verified that the most general solution of this partial dififerential equation has the form 20. In a manner entirely analogous to that of Art. 18, it may be seen that the linear partial differential equation of the first order in n variables, ^.|+^>|+-+^»a¥.=» (') where the Xj, ..., X„ are certain functions of x-^, ..., «„, SIMULTANEOUS SYSTEMS. 19 represents the same problem as does the simultaneous system of ordinary differential equations dxi_dx^ dx„ x;-x;--=x: ^^^ Also it follows from considerations similar to those of Arts. 17 and 18, that (7) cannot have more than n—l independent solutions. If these are of the form . Ui(Xi, ..., Xn), Ui{Xi, ..., X„), .... Un-liXi, ..., X„), the U'b being put equal to arbitrary constants, will give the integrals of (8). Moreover the most general solution of (7), or the most general integral function of (8), has the form ^(U„ U^,..., Un-l). SECTION III. Integration of Ordinary Differential Equations in Two Variables, in which the Variables can be separated by Inspection ; and of a Special Form of a Simul- taneous System in Three Variables. 21. Although we are not yet ready to present any general theory of integration of ordinary differential equations, it will be necessary for us to call attention here to the fact that when the variables can be separated by inspection in an ordinary differential equation of the first order in two variables, so that the equation may be written X{x)-Y{yy ^^^ its complete integration, which is virtually a problem of the Integral Calculus, may be immediately accom- 20 ORDINARY DIFFERENTIAL EQUATIONS. plished. The general integral will have the form dy ]X(x) J] jz(^-jF(2/r'°°'*" ^^^ and (2) is considered the general integral of (1), whether the functions in equation (2) can be expressed in a form free from the sign of integration or not. Of course the differential equations in which the variables may be separated by inspection constitute only a very small class of all ordinary differential equations of the first order in two variables ; but we shall see that the integration of these, the simplest possible differential equations of the first order, will, in a future chapter, furnish us with the means of integrating whole classes of very complicated equations. Example 1. The ordinary equation in two variables {l+x)ydx + {\ -y)xdy=0 may be written -dx-\ ^y = 0. X y The general integral will therefore have the form j-^'^^ + /-^y = const. , which is seen to be log(xy) + x—y = const. The given ordinary differential equation is, moreover, equivalent to the linear partial differential equation (i-y)-g-(i+-)y|=o; and it may at once be verified that if ^og(xy)+x-y, or any function of this function, be put in place of / in the linear partial differential equation, that equation will be satisfied identically. Example 2. Given the equation dx _^ dy ^^ •Jl-x' -Jl-y^ SIMULTANEOUS SYSTEMS. 21 Here the variables are already separate, and the general integral is, 8in-'a;+sin~'y=a. (a = con8t.) But a function of an arbitrary constant is itself an arbitrary con- stant : hence, taking the sine of both members of the last equation, and replacing sino by c, we see that the general integral may be written Wl— _y^+yVl — 3? = c. (c = const.) It may be readily verified that any function of the integral function Wl — 1/^ + 7/s/l -x' is a solution of the linear partial differential equation 22. Similarly, if a given simultaneous system in three variables has the very special form dx _ dy _ dz x{^-iW)-wy its integrals may also at once be written in the forms { dx { dy , C dy [ dz Example. The simultaneous system dx_dy_dz X y z evidently has for its integrals log X - log y = const., log y - log z = const. ; or, as they may be written, - = a, - = 6. (a, h const.) y ' z It may at once be verified that any function #(-, 'M is a solution of the linear partial differential equation, equivalent to the above simultaneous system, 22 ORDINARY DIFFERENTIAL EQUATIONS. 23. It may, finally, be noticed that if the given simultaneous system has the particular form dx _ dy _ dz X^^)-Y{^)~ Z{x, y, z)' and if the integral of the ordinary differential equation in two variables, dx _ dy has been found, either by separating the variables, or by methods to be explained later, in the form U{x,y) = c, (c = const.) then the last equation may be used to eliminate either of the variables x or y, as may be desired for the purpose of integration, from X, F, or Z. If, for instance, we find from the last equation y = (c,x), the second integral of the given simultaneous system may be found by integrating an ordinary differential equation in two variables of the form dx _ dz X(x, 4>)~ Z(x, 4,, z)' where, of course, the value of in terms of x and c has been substituted in place of y. If the integral of this equation has been found in the form W(x, z, c) = h, (b = const.) we now substitute for c its value U(x, y), finding the second integral required in the form V(x,y,z) = b. The reader will bear in mind that the above is only a very special form of simultaneous system in three SIMULTANEOUS SYSTEMS. 23 variables. A general theory of integration of such differential equations will be given later ; but it is convenient to notice these simplest forms now, in order to make use of them in the next chapter. Example. Given the simultaneous system dx _dy _dz x'^~ xy~ An integral of is found to be Hence, in the equation we may put for x, -. = c. z' dx _dy .ifi~ xy 1-. X dy _dz xy z^ Thus we find cdy _dz (c = const.) y (6= con St.) of which the integral is Now put for c its value, -, and we find as the second integral 1_1_ Z X X' required -\ x-z or = 6. Of course this result might have been obtained directly from dx dz without any intermediate steps. It may readily be verified that any function of the form Kl? xz ) is a solution of the linear partial diflFerential equation which is equivalent to the given simultaneous system. 24 ORDINARY DIFFERENTIAL EQUATIONS. EXAMPLES. Integrate the following ordiuary differential equations of the first order in which the variables may be separated by inspection, giving in each case the equivalent linear partial differential equation in two variables, and verifying that the integral-function of the ordinary equation is a solution of the linear partial equation : (1) % = r.y^.. (2) J^ + ^^ = o. (3) {y^ + xy^)dx+{x'^-y3p)dy = 0. . xdx ^ ydy ^ ' l+y l+x (5) sinx COB ydx= cos X sin y dy. (6) {\^y'^)dx={y+>JY+f){\+3F)^dy. (7) &%c^xta,uydy + sech/\,a,'axdx=0. Give the linear partial differential equations equivalent to the following simultaneous systems ; integrate the simultaneous systems, and show that any function of the integral functions of each simultaneous system is a solution of the corresponding linear partial equation. ,g. dx^dy^dz^ ,j^. dx^di^dz ' X y —z yz xz xy . dx dy_dz . dx_dy_dz^ (^n\ dx _dy dz_ ^^^>~y-~^~l+z^- In (12) the symbol -r- is used merely to show that the coefficient of ^ in the linear partial differential equation equivalent to (12) is Ox zero. That partial differential equation is and since J- does not occur at all, it is clear that .j; is a solution of the equation : that is, .r= const, is one integral of (12). CHAPTER III. THE FUNDAMENTAL THEOREMS OF LIE'S THEORY OF THE GROUP OF ONE PARAMETER. 24. We propose to develop in the present chapter such of the propositions which Lie has established with reference to the transformation group of one parameter, as we shall need subsequently in the integration of ordinary differential equations. The theory of the group of one parameter in two variables is minutely explained in order to enable the reader to make use of the group as an instrument for investigation. For the sake of greater clearness, we shall generally limit ourselves to two variables in establishing the necessary fundamental propositions ; but the method of extending the results to n variables, in such cases as it is desirable, will be sufficiently indicated. SECTION L Finite and Infinitesimal Transformations in the Plane. The Group of one Parameter. 25. By a transformation of the points of the plane, we understand an operation by m^ans of which every point of the plane is conveyed to the position of some point of the same plane. 26 ORDINARY DIFFERENTIAL EQUATIONS. The general form of a transformation of the points of the plane is given by the system of equations x^ = {x,y), yi = \lr(x,y), (1) where ^ and -^ are independent functions of x and y. We suppose here that the coordinate axes remain un- changed ; but every point of general position (a;, y) is conveyed to a new position of which the coordinates are {x, y, a), y^ = \lr(x, y, a), (3) where a is a parameter which can assume oo^ continuous values. In general, then, it will not be the case that the performance of any two transformations of the family (3) successively upon the points of the plane will be equivalent to the performance of a third transforma- tion of the family (3) upon those points. For instance, the equations a;i = a-a;, y^ = y represent a family of transformations which do not possess the above peculiarity. For if a;2 = ai-a!i, 2/2 = ^1 be a second transformation of the family, we find, when THE GROUP OF ONE PARAMETER. 27 the two transformations are successively performei upon the point {x, y), that this point assumes a position given by But the transformation given by the last equations does- not belong to the original family, of the general form^ ajj = const. — X, 2/i = 2/. If, now, x^ = (t>{x, y, a), 2/1 = 1/' {x, y, a) be any given transformation of the family (3), and if x^ = 4>{x^, 2/1, Oi), 2/2 = i^{^v Vv «i) be a second transformation of that family, then the transformation which results from performing these two successively evidently has the form *2= ^{0(«. y> «). V'C^. V' «). "J. 2/2 = V'{^(^. y, a), ^{^, y, a), aj. If it happens that the right-hand members of these- equations have the general forms {{x, y, ao), y = \}r{x, y, %) ; If, now, we assign to the parameter a a value which ■differs from a^ only by an infinitesimal quantity, say Of^ + Sa, the corresponding transformation Xi = , y, ao+^«). 2/i = V'(a;. y, ao+^*) (^') will differ only infinitesimally from the identical trans- formation ; that is, (6') will be an infinitesimal transfor- mation. By Taylor's theorem. ,-4,{x,y,a,)+ ^^ da+ ^-^ 1.2+-' y^-xlr(x,y,a,)+ ^^^ Sa + —^ 172+-' or, from the above value of the identical transformation, ^1-^+ 3^; ^'^^ 3< 172+-' y^-y+ — 3^; — ^""^""av — t:2+-- Thus we see that Xj, j/j really differ from x and y by infinitesimal quantities. If the coefficients of all powers of Sa up to the r**" vanish for all values of x and y in the last equations, we introduce St = Sa^ as a new infinitesimal quantity, and so obtain the equations of the infinitesimal transforma- tion in the general form x^=x+^(x,y)St+..., yy = y+t,{x,y)8t+.... Here ^ and ;; also contain a^; but since a^ is a mere number, it is not necessary to write it explicitly in ^ and Tj. THE GROUP OF ONE PARAMETER. 31 It is true that by this method for finding the infin- itesimal transformation of a given G^ (6), it is impossible to say whether the succeeding terms of the last equations involve integral or fractional powers of St ; this difficulty is however avoided by a second method given below. 28. Let a fixed value e be assigned to the parameter a in the Cj, ^i = <^(^> y, as). yi = 'f (^, y. a), (6) and suppose that the corresponding transformation, which we shall designate as the transformation (e), carries the point of general position P to the new position P^. Then, by hypothesis, the trans- formation in the G^ (6) which is inverse to (e) will carry the point Pj back to the position P. Now if the parameter of the last trans- formation be designated by e, it is clear that a transformation with the parameter e-t-8e, where 8e is an infinitesimal quantity, will carry the point P-^ not exactly back to P, but to a position P which is at an infinitesimal distance from P. If the transformations (e) and (e-t-Se)be performed successively, the result must be equivalent to the performance of a third transformation of the family (6) ; one that will take the point P directly to the position P". But since the distance PP is infinitesimal, the transformation which carries the point P directly to the position P is called an infinitesimal transformation. The above geometrical considerations may be carried out analyti- cally. The first transformation is represented by Xi=4,{x, y, e), yi=^(x, y, e) ; and the second by xf = <^(xi, yi, e + 6e), y'=f(xi, y^, e + Se), where we suppose (x, y), (xj, yj, and (y, y') to be the coordinates of the three points P, P^, and P respectively. The transformation which carries P directly to F is found by eliminating x^, y^ from the above equations : we find x'={{x,7j,e), ir(x,y,e), e+Se}, y'=f{^{x,y,e), f(x,y,e), e+Se) Developing in powers of Se, we have x' = {(f>(x,y,e), yfr{x,y,e), e} + g| be+..., y'=f{(x,y,e), ylr{x,y,e), e} + g| beJr..., 32 ORDINARY DIFFERENTIAL EQUATIONS. But since the transformations (e) and (e) are inverse, we have the identities, x = ^{{x, y, e), ir{x, y, e), e], y = ir{{x, y, e), ^{x, y, e), e] ; and the last two equations become y__^. I 3<^{<^(^. y, e),jr(x, y, e), e} g^ ^ , .. . 'dj^Wx, y, e), ir{x, y, e), e }^_ , . y =y+ 91 Se+... , nd it is evident that these equations represent an infinitedmal transformation. It is easy to see that the coefficients of 8e above do not vanish identically ; for they may be written respectively : and if these expressions were identically zero, the equations (6) would necessarily be free of any parameter, which is contrary to hypothesis. Since e depends upon e alone, the equations to the infinitesimal transformation may evidently be written 3f=x+^(x,y, e)Se + ..., y'=y+-n{x,y,e)Be+,..; and it is clear that every Oi in the plane contains at least one infinitesimal transformation. 29. If < be a parameter, it follows from the last two articles that the general form of an infinitesimal trans- formation in two variables will be y^ = y+r,(x, y)St+..J '^ '' We shall, as usual, neglect higher powers than the first of the infinitesimal quantity St; and hence the increments which x and y receive by means of the above infinitesimal transformation have the forms Sx^i(x,y)St, Sy = r,(x,y)St. (8) THE GROUP OF ONE PARAMETER. 33 It is clear that this transformation assigns to every point {x, y) of general position, a direction through which it is to be moved, given by Sx i(x,y)' and also a distance through which it is to be moved, given by >/Sx^+Sy^ = n/^+V. St. y As far as determining a direction through which a point of general position is to be moved is concerned, the infinitesimal transformation offers an analogy to the ordinary differential equation of the first order in two variables (Chap. I, Sec. II). We can get a clear and fruitful idea of an infinitesimal transformation, if we suppose that we put aU the points of the plane into motion simultaneously, by performing upon them the infinitesimal transformation (8) an in- finite number of times. In this manner a point (x, y) will assume a simply infinite number of continuous positions, which form a curve. The whole change of position of the points of the plane, since it is repeated from moment to moment, may be called a permanent motion, and may be compared to the flow of the molecules of a compressible fluid. If t represents the time, and we measure it from a fixed point, say t = Q it is clear that] the point of general p.c. 34 ORDINARY DIFFERENTIAL EQUATIONS. position {x, y) will, after the time t, arrive at a new position (ajj, j/j), where the coordinates x^, y^, are functions of X, y, and t. If t increases by dt*, oc^ and 2/^ will, by (8), receive the increments dxj^ = iix^, y^)dt, dy.^ = r,(x^, y^dt, 80 that asj and j/j may be found as functions of t by integrating the simultaneous system £K, 3/i) "zK. 2/1) The first of these equations has, as we know, an integral of the form U{x-^, i/j) = const., and by Art. 23, the second equation has for general integral, V{x^, 3/j) — t = const. Since at the time t = the point (ajj, j/j) must be at the fixed position {x, y), we must choose the arbitrary constants in the last equations in the forms U{x, y), V{x, y) ; so that ajj, 2/1 are given as functions of t, x, and y, by the equations ^K3/i)=T^(a3, 3/) \t} <"> These equations obviously represent a Ctj, with the parameter t ; and that such must be the case was clear, a priori, from the kinematic illustration. For, if in the time t the permanent motion carries the point {x, y) to the position (x-^, y-^) — and in the time t.^ carries the point (x^, 2/j) to the position (x^, y^) — it is evident that in the time t-\-\ the point {x, y) will be carried to the position * We may clearly use either of the symbols S or d, to indicate an infinitesimal increment. Here we make use of c2 in order that the simultaneous system may appear in the usual form. THE GROUP OF ONE PARAMETER. 35 (^2' 2/2) ; tl^^t is to say, the successive performance of any two transformations of the family (9), with the values t and {x,y,t), y^ = yjr{x,y,t) (10) be the finite equations to a G-^, we can evidently consider any function of the form /(aJj, y^ as a function of x, y, and t ; and for that value of t which gives the identical transformation, say for ^ = 0, we must have x.^ = x, y^^y, and hence f{x^, y^) =f(x, y). Since /(a;,, t/j) varies when t varies, we are led to inquire as to what increinent, Sf, the function f(x^, y^) receives, when x-^ and j/j receive their respective increments, &i = ^(fl^i, y-^)St, Sy^ = ,,(x.^, y^)8t. We find Sfi' and the law of formation of the coefficients in the expansion (11) is now obvious. If we put ^ = in the coefficients of (11), then x-^ and i/i are changed into x and y ; also UJ becomes Uf; U^(UJ) becomes U(Uf), etc. Thus we arrive at the important expansion f{x^,yd=fi<^>y)+{uf+^u(m+ (12) This holds, of course, when /j has the particular values Xj, and j/j. Thus x,=.x+*^U(x) + ^UiUix))+. \ (13) y^ = y + LU{y)+^U{U{y})+. and these are evidently the finite equations of the G^ of which is the infinitesimal transformation. The equations (13) are of course only another form of the finite equations THE GROUP OF ONE PARAMETER. 41 found, Art. 29, by integrating a simultaneous system. The reader may readily see that the results of this Article may at once be extended, mutatis mutandis, to n variables. Example 1. Suppose the infinitesimal transformation is given ; and we wish to find the finite equations to the 0^ Here it is seen at once, Art. 34, U(^) = -y. U(j/) = X, U{U{x)) =-^, U{U(y)) =-y, U{U{U{x))) ^ y, U{U{U{y))) ^-x, U{U{U{U{x))))^ X, U{U{U{V(y))))^ y. Thus, by (13), ^>=^-i2'-o^+r:2:3^+iT2r3T4^- -' By well-known developments of the Differential Calculus, the last equations may be written Xi=xcost—yamt, yi=x bid t+y cost. Hence the O^ is the G^ of rotations, mentioned Art. 26. Example 2. Given ^f-4x-4 to find the finite equations of the 6i. Here, proceeding as above, we find the expansions t f^ Xi=X + -X+:7—^X+...=Xef, t t^ 42 ORDINARY DIFFERENTIAL EQUATIONS. Instead of e* we may choose a as the parameter of the G^, and we find as the finite equations, In a number of the most important 0{& it will be found that all the terms in the series (13), after the second, are zero. SECTION II. Invariance of Functions, Gv/rves, and EqiMtions. 37. Suppose, now, that we demand that a given function of x and y, of the form Q(x, y), shall be invariant when we perform upon it the transformations of a given 0^. That is, if -the infinitesimal transforma- tion of the given ffj be Uf^iix.y)^+r,(x,y)^^, and the equations to the finite transformations be a'i = 0(«. 3/. 0. yi = ^{«!,y,i), (1) we demand that when, by means of (1), Q is expressed as a function of x^, y^, Q must be the same function of cCj, 2/i that it was of x, y. Thus we must have, for all values of t, ^(a;i. 3/i) = ^(aJ. 2/). by means of (1). But, from (12) in Sec. I., the last equation may be written ^(x, 2/)+| f7(fi)+^Cr(Cr(Q))+ ... = Q(x, y); and we see that a necessary and sufficient condition that Q,(x, y) shall be invariant under the 0^ (1) is that C/"(fi)=0 (2) If this condition be fulfilled, fi is called an invariant of the Gj (1). INVARIANCE. 43 The condition (2) may be written out in full and this shows that Q is a solution of the linear partial differential equation in two variables or an integral-function of the equivalent ordinary differ- ential equation ckc _dy Hence we see that, by Art. 17, a (?i in two variables always has one invariant; and every invariant can be expressed as a function of any one invariant. Example. The function is an invariant of the O-^ of rotations ; ail =x cos t—ysiw t, yi=x sin t +y cos t. For, from the last equations, y=yx cos t—x^ sin t ; hence Q,{x, y) = x'^+y^={xi cos t+y^ sin tf+{yi co&t-Xi sin tf = x-^ (cos^ t + sin2 1) -Hyi''(sin2 1 + cos^ t) Hence fl has the same form in the variables x^, y^, for all values of t, that it has in the variables x, y ; i.e., J2 is an invariant of the (?,. The infinitesimal transformation of this (?i is and we may at once verify the fact that ?7(I2) = ; for 44 ORDINARY DIFFERENTIAL EQUATIONS. We see that the verification of the fact that Ji is an invariant is much simpler when accomplished by means of the infinitesimal transformation of the (?i, than when accomplished by means of the finite transformations. 38. Every point of general position in the plane describes, Art. 29, a continuous curve when the infini- tesimal transformation of a given G-^ is performed upon it an infinite number of times. We shall call this curve the path-curve of the point under the transformations of the Gj ; and it is obvious that each Gj may be said to have x^ path-curves, one through each point of general position in the plane. The direction through which a point (x, y) is moved by a given (?j, of which the infinitesimal transfonnation is is given, Art. 29, by Sy^ r,{x,y) Sx i(x, y)' Now if Q,(x, y) be an invariant of the G-^, we saw that Q, must satisfy the linear partial differential equation But this partial differential equation is equivalent to the ordinary differential equation #(«. y)dy-ri{x, y)dx = 0. That is, fl must be an integral function of the last equation ; and the integral curves il(x, 2/) = const. have in each point the tangential direction dy^ T,{x,y) dx ^{x, y) INVARIANCE. 45 Hence an invariant, Q,{x, y), of a G-^ in the plane, being written equal to an arbitrary constant, will represent that family of oo^ curves in the plane which we call the path-curves of the Gy 39. It should be noticed that any point, or points, in the plane for which i{x,y) = r,{x,y) = 0, are absolutely invariant under the infinitesimal trans- formation of the given G-^, since in these points x and y do not receive any in- crements at all. Example 1. The infinitesimal transformation of the O^ of rotations is Thus, as we know, the invariant must be a solution of the linear partial difierential equation 46 ORDINARY DIFFERENTIAL EQUATIONS. That is, iJ is the integral-function of dx _dy -y~ ^' or of xdx-\-ydy = l}. The integral-function of this ordinary differential equation may obviously be assumed to be Q, = x'+y\ Hence the path-curves of the G^, that is, the curves which the points of the plane describe when they are subjected to the trans- formations of the Oi of rotations around the origin, are the circles ii = .r^ -I- y^ = const. This was, of course, geometrically evident a priori. The origin is obviously an absolutely invariant point. Example 2. Suppose the infinitesimal transformation to be given. T The invariant is found as the solution of 312 , 3n „ or as the integral-function of xdy-ydx=0. This integral-function is obviously fl = -. Hence the path-curves INVARIANCE. 47 of the (?i, of which x^^ + xy^J- is the infinitesimal transformation, are the straight lines through the origin V -= const. X The absolutely invariant points are given by x'^=xy = 0, that is, x=Q. Thus the ^/-axis is an invariant straight line, which consists of absolutely invariant points. 40. A family of oo^ curves in the plane, considered as a whole, may be invariant under the transformations of a given (tj in two ways ; each curve of the family may be separately invariant, when, of course, the family is, as a whole, also invariant ; or the curves of ike family may, by Tneans of the transformations of the 0^, be inter- changed araong each other, leaving the curve-family as a whole, however, still invariant. We have seen that the path-curves of a given 0^ are a family of oo^ curves which is invariant in the first way, that is, each member of the family is separately invariant. Usually, however, when a family of oo^ curves in the plane is invariant under the transformations of a given (r^, the individual members of the family are not in- variants, but are merely interchanged by means of the transformations of the 0^ Let f2(a;, 2/) = const. be any family of curves in the plane, which, as a family, are invariant under a G-^ whose finite transformations are given by the equations a^i = ix, y, t), yi = ^(x, y, t), whilst the infinitesimal transformation of the 0^ is TJf^i{x,y)%+r,i.,y)^ Since the curve-family is to be invariant, the equation 48 ORDINARY DIFFERENTIAL EQUATIONS. to the curves must, in the variables ajj, y^, have a functional form either identical with, or equivalent to, that in x and y; that is, the equation to the invariant family may be written, in the new variables, in the form ^(''^i' 2/1) = const. Now we know that ^{p,{x, 3/)) = const, represents the same family of curves that Q,{x, y) = const, does ; hence we may write, as the condition that the family Q,{x, y) = const, shall be invariant. If the left-hand member of this equation be developed by means of (12) in Sec. I, we find that a necessary and sufficient condition that the curve-family Q.{x, 1/) = const, shall be invariant, is that U{^(x,y))=-F{^{x,y)). When a relation of this form holds, we sometimes say that the family of curves admits of the transformations of the Gj. For the partictdar case that F(fi(a;, i/)) = 0, the above condition gives, as it should, the family of invariant path-curves. Example 1. We saw that the concentric circles, Art. 39, j;2-f-y2=^2 (r = const.) are the path-curves of the O^ of rotations ; and hence, of course, they form a family of curves which are invariant under that O^ in such manner that each curve is separately invariant. But the family of oc^ circles is also invariant under the G^, x^=xt, y-^=yt, with the infinitesimal transformation For, from the above equations, X, y. INVARIANCE. 49' and substituting tliese values in the equation to the circles, we find or x^+y^ = cons\,. ; which is an equation of the same functional form in x^, y, that the original equation was in x, y. Thus, by means of the finite transformations of the (rj, we see that the curve-family as a whole is invariant, while the individual members are obviously not invariant. We may at once verify the same thing by means of the infinitesimal transformation Uf. For here U(Si)='0{x'^-Vy'^) = ^x.x-\-%y.y = '2{3fi-Vy'^). In this case, therefore, or the curve-family is invariant. Example 2. The family of straight lines V - = const. X admit of the G^ of rotations around the origin. This may be readily verified by means of the finite equations of the rotations. But the infinitesimal transformation is and since in this case 12 = -, we find = 3^+1 = 122-1-1. x' Hence the condition that £/'(I2) = F(12) holds in this case. 41. The results of Arts. 37-40 may be readily extended, mutatis Tnutandis, to three or more variables. P.C. D 50 ORDINARY DIFFERENTIAL EQUATIONS. Thus, if Uf^iix, y, ^)%+r,{x, y, ^)^+ti^, y, ^)|{ be the infinitesimal transformation of a (tj in three variables, the points, or curves, for which are absolutely invariant under the G■^^. Also, the necessary and sufficient condition that a family of oo^ surfaces, Q,{x, y, «) = const., shall be invariant under the G-^ is that C/"(fi) = F(f}). 42. In a manner entirely analogous to that of Art. 37 it is seen that the necessary and sufficient condition that an equation of the form n(x,y) = shall be invariant under a given (?i, Uf, is that the expression U(Q) shall be zero, either identically or by means of fi = 0. This condition may at once be extended to n variables. Example 1. The equation Q = x^+y^-l=0 is invariant under the &'„ For here 7T/<->\ 3fi . 3ii ^ ;, „ C^W= -2/.^ + x~=-2x7/ + 2x7/=0. Hence the condition for an invariant equation is satisfied. Example 2. The equation n=:i/-x = is invariant under For here ^("y-^dx'^^dj- -'^+y-^- Hence the condition is satisfied. INVARIANCE. 51 43. We shall now find all equations of the general form fi = 0, which are invariant under, or " admit of," a given G.^, Uf: and as this result is very important for future use in more than three variables, we shall develop it at once in n variables. If the given (?j, in the n variables x^, ..., Xn, have the form UJ = ^i{Xj , ..., ^n)^ + • • • + fn(^i, • • • , ^»)^ ' it might be possible that the ^j, ..., ^„, are such function that they all become zero by means of an equation which is invariant under the G^. If we represent the equation by \l\X-^, . . . , Xn) — V, it is true that in this case Q = is an invariant equation ; but the system of values of the variables which satisfy fj = is not transformed at all. For instance, in two variables, the equation is evidently invariant under the G^, inasmuch as the infinitesimal transformation of the (tj vanishes entirely when x^-'ry^—l is zero; and the (?j does not transform at all the system of values of X and y, which satisfy the equation We shall, in future, exclude from consideration an invariant equation which makes all the ^j, ..., ^„ iden- tically zero. Thus we may assume that one at least of the ^■^, ...,£„ in Uf does not become zero by means of the equation £2 = 0. Let us assume that ^„ is not zero; then, by 52 ORDINARY DIFFERENTIAL EQUATIONS. Art. 42, it is clear that if Q = is invariant under the infinitesimal transformation Uf, it will also be invariant under the transformation For, if U{Q) is zero, either identically, or by means of f2 = 0, it is clear that F([2), which is U('[l) divided by |„, will also be zero, either identically or by means of ^ = 0. Now the linear partial differential equation of the first order in n variables, 17=0, has (n — l) independent solutions which are functions of x^, ..., x„, and which we shall designate as Vl, 3/2. •••,2/n-l- But if we consider a;„ in connection with these (w— 1) independent functions, it is clear that the n functions Vv Vi' •••' y^-it Xn must also be independent. Otherwise we might express x„ as a function of i/j, ..., j/n-i, say in the form Xn= y^iVv ■■■,yn-i)- But, Art. 20, the last equation means that a;„ must be a solution of the linear partial equation Yf= ; which is manifestly impossible, since for f=Xn this equation reduces to 1=0. Hence the n functions y■^, ..., j/n-i. Xn are independent, and we may introduce them as n new independent variables. By Art. 35, it will be easily seen that ly then assumes the form ^, dXn which is a mere translation. Hence, we may remark, incidentally, that by a proper INVARIANCE. 53 choice of variables, every infinitesimal transformation may be brought to the form of a mere translation. In the new variables the equation f2 = has the form F(2/i, •.•,2/n-i, a;„) = 0, and Xn can only occur formally in this equation. For if Xn be really present, we might solve and find a;„ in terms ^^ 2/1. ■■•. 2/n-i. SO that the invariant equation will have the form F = a;„-*(2/i 2/n-i) = 0. But for this equation to be invariant under Yf, we must have F(F) zero, either identically or by means of F = 0. Now F(F)=y(ic»-*)=l; and hence we see that the variable Xn cannot occur in the function F. If now we return to our original variables and desig- nate the equation which is invariant under Uf by f2 = 0, it is clear that Q must be capable of being expressed as a function of the (n—1) independent solutions VvVv •• -.2/71-1 of the linear partial differential equation Yf=0, or of its equivalent equation Uf= 0. This is a result of much importance for our subsequent investigations. For the special case of three variables, it follows that to find the most general equation which is invariant under a given G^, Uf=i{x, y, z)%+n{x, y, ^)^+f('^' V' ^)%' it will be necessary to find two independent solutions of the linear partial differential equation of the first order 54 ORDINARY DIFFERENTIAL EQUATIONS. If these solutions be u{x, y, z) and v(x, y, z), the most general invariant equation will have the form or, written in a form solved for u, u=f{v). SECTION III. The Lineal Element. The Extended Group of One Parameter. 44. A lineal element is the aggregate of a point {x, y) in the plane, and a direction through that point. If y' represents the tangent of the angle which the direction makes with the a;-axis, it is clear that x, y, y' may be regarded as the coordinates of the lineal element ; and by assigning to y', which need not necessarily be con- sidered a differential coefficient, all possible numerical values, we evidently obtain the x^ lineal elements which pass through the point {x, y). An ordinary differential equation of the first order in two variables, of the form ^(«. 2/. 2/') = 0, may now be considered as an algebraic equation in the three variables x, y, y , defining oo^ of the qo* lineal elements of the plane. The equation Q = 0, as a differ- ential equation, has 00^ integral curves; and the tangent to an integral curve at any point (x, y) must be determined by a value of y' which satisfies the above equation. But the same value of y' determines the lineal element through the point {x, y) ; for when x and y are fixed, only that value of y' will satisfy Q. = 0. Thus the x^ lineal elements which are defined by the algebraic equation in three variables, fi = 0, envelope the integral curves of THE LINEAL ELEMENT. 55 the differential equation in two variables, Q = 0, as indi- cated in Fig. i. Fig. 1. If the equation Q = happens not to contain y' at all, it still represents oo* lineal elements, although it can no longer be considered a differential equation. These are Fig. 2. evidently the oo^ lineal elements whose points lie along the curve = 0, as indicated in Fig. 2. Through each point pass oo^ lineal elements, since at that point x and y are fixed, while y', being indeterminate, may have x^ different values. In the following, as we have only to do with differ- 56 ORDINARY DIFFERENTIAL EQUATIONS. ential equations, we shall always consider that Q actually contains y'. 45. If a transformation be given by the equations ^i = {x,y), yi = \j^ix,y), (1) it is obvious that not only the points of the plane, but also the oo^ lineal elements are transformed by (1) according to a fixed law. For the value of the trans- formed y', which we shall call i/\, and which determines the direction of the transformed lineal element, is determined by means of the equations, 'dx dy ' ^ Thus it is seen that the value of y\ depends merely upon the transformation (1) and the values assigned to X, y, and y'. The transformation in the three variables X, y, and y', Xi = {x, y, b), y^ = \J^(x, y, b), (5) b being a function of a and a^ alone. If each of the transformations (3), (4), and (5) be extended, it is easy to see that all the extended trans- formations form a Gy For (3), when extended, becomes x, = ^ix, y, a), y, = i.{x, y, a), j;^ ^^||-^'^«) ; ...(6) and (4) becomes The successive performance of (6) and (7) upon the lineal elements of the plane is equivalent to the per- formance upon them of the transformation obtained by eliminating x-^, y-^ between (6) and (7). But by (5), the latter transformation must have the form x^ = ,p(x,y,b), y2 = ^/r{x,y.b), y'2 = g^g ^' ^j (8) where 6 is a function of a and a■^ alone. It is clear that (8) is the transformation which would be obtained by extending (5); that is, the oo^ extended transformations, corresponding to the G^ (3), form themselves a G^ 47. It is also obvious that if a point transformation of the form (1) be given, not only will y', but also y", ..., yf"), be transformed by (1) according to fixed laws. The transformation in four variables, x^ = (j>{x,y), yi = i^(x,y), 3/i = ^. V 1=-^^' is called the twice-extended transformation corresponding to (1). Each of the oo^ transformations of the G^ (1) may be twice- extended in this manner; and it is very easy to see that the «i twice-extended transformations in the four variables x, y, y', y" also form a G^ 58 ORDINARY DIFFERENTIAL EQUATIONS. Similarly, it may be shown that the thrice- extended transformations of a given Gj in the variables x, y, y', y" , y" form a (tj, and so on to the ■n-times extended trans- formations of the ffj. 48. We shall now give a method for finding the in- finitesimal transformation of an extended ffj, since the conditions for the existence of such a transformation, Art. 26, are obviously fulfilled. If the finite transformations of the G-^ be given by the equations x^ = =.+ .4l=, y.=y^j4^,; uf^-^lxf+y^). •Jx'+y' •^Jx'+y^ 'Jx^+y^^ ox Oy ' By means of the finite transformations of this Gj, all points are moved along their radii vectores through the same distance t. (6) x^ = tx, y^ = ~y ; Uf=x^-y^. (7) Xi=^xcost — ys\r\t,yy=xs,\i\t+ycost; Uf=—y^-'rxJ-- (8) -.=^, .v.=,-^ ; Uf^ 4A..^|. (10) Show that the family of all oo" conic sections whose axes coincide with the coordinate axes, is invariant under the Oy of afiBne transformations, Xy = tx, yy=^y. Verify the result by making use of the condition. Art. 40, t^((i)) = I2((i)). (11) Show that the family of oo' concentric circles, x^+y'^=r'^, is invariant under the G^ of similitudinous transformations, Xi = tx, yj, = ty, and verify the result by Art. 40. EXAMPLES. 61 (12) Show that the family of oo^ straight lines, X y , a b is invariant under each of the O^ given in examples (l)-(4) and (6)-(9) ; that is, that these G^ are projective. Verify the results, as usual, by Art. 40. (13) (a) Show that the family of oo^ circles with radius 1, (x-a)2 + (y-6)2 = l, is invariant under the G^ of rotations given in example (7). (6) Show the same of the family of oo' tangents to the circle, «-2 + a^ = l. [See Ex. (17), Chapter I.] (14) Show that the family of oo^ circles, (,x-af+f' = \, is invariant under the G^ of translations, XT^=x + t, yi=y. (15) Show that the family of oo' circles which touch both axes of coordinates, is invariant under the G^, Xi = tx, yy=ty, verifying as usual. CHAPTER IV. CONNECTION BETWEEN EULER'S INTEGRATING FACTOR AHI> LIE'S INFINITESIMAL TRANSFORMATION. 50. We are now prepared to show to what the develop- ments of the preceding Chapters have been tending. In the first section of this Chapter, we shall show how to integrate the exact differential equation of the first order in two variables. In the second section, we shall show that a differential equation of the first order in two variables which is invariant under a known G^ may always be integrated by a quadrature ; while in the third section, we shall establish some of the most important types of such invariant equations. SECTION I. Exact EqvMtions of the First Order. Integrating Factors. 51. A differential equation of the form di^{x,y) = —dx+—dy = 0, (1) since it is obtained by the complete differentiation of an equation of the form ^{x, 2/) = const., EXACT EQUATIONS OF FIRST ORDER. 63 it is said to be an exact differential equation; and the first member of (1) is called a complete differential. It is obvious that not every differential equation of the first order, X{x,y)dy-Y(x,y)dx = 0, (2) is exact ; for, to be exact, it is necessary that the condition X=— F=- — dy' " dx be fulfilled. But from this follows dX^_dY ,„. dx dy' ^ ^ since each of these quantities must be an expression for "dydx We shall see that this necessary condition that (2) shall be an exact equation is also sufficient. For the most general function, #, which satisfies i--F(..,), is obtained from * ^-^Y(x,y)dx+Z{y); the integration being performed as if y were a constant, and Z being a function of y alone, which occupies the place of the constant of integration. The only other condition to be satisfied is that the partial differential of # with respect to y shall be equal to X(x, y) ; that is, X{x, 3/)^-^{-JF(a;, y)dx+Ziy)}, (4) f^^X{x,y)+^jY(x,y)dx (5) Since Z is free of x, the second member of this identity 64 ORDINARY DIFFERENTIAL EQUATIONS. must also be free of x; that is, its partial differential with respect to x must be zero. Hence ■dx^-dy ' dx 'dy which is exactly the condition (3); that is, (3) is a necessary and a sufficient condition that the differential equation (2) shall he exact. From the above it follows that the integral of the exact equation (2) may be found by quadrature in the form *- -\y{x, y)dx + \{x{x, 3/) + W|L^)d2/ = const.; or, if more convenient, the equivalent formula, *^ -^X{x, y)dy+^(Y{x, y)+'^&^^)dx = couBt., may be used. Here the integration with respect to y is to be performed as if a; were a constant; and with respect to tc as if 2/ were a constant. It may be remarked that the equations of Chap. II., Sec. II., are a special class of exact equations. Example 1 . In the case of the differential equation (y2 — ixy — 'i,3^)dy + {3^ — ixy - 'iy^)dx — 0, the condition (3) is satisfied. For A'=/-4.^2/-2^^ r=-(^-4i2^-2y2), whence, as may be at once verified, so that the diflferential equation is exact. EXACT EQUATIONS OF FIRST ORDER. 65 Using the first of the above formulae for $, we find r a? - J ^{^> y) may be used advantageously in this case. We have -Ixdy^-yl; thus 'b\Xdy _ y2 'dx ~ x^' Hence and the integral of the last expression with respect to x is therefore X. Hence the general integral sought is - — + ^= const., X or x^-y^=cx. (c = const.) p.c. E 66 ORDINARY DIFFERENTIAL EQUATIONS. 52. It will usually not be the case that the functions X and Y in (2) satisfy the condition (3). But since every diflPerential equation of the first order of the form (2) must have an integral of the general form Q,{x, 2/) = const., (6) the equation dx By * must be equivalent to (2). That is, there must always exist a function M(x, y), such that we can write ^dx + ^^dy^M{x, y){Xdy-Ydx); and since the left-hand member of this identity is a complete differential, the right-hand member must be a complete differential also. From (3) we see that M, X, and Y must satisfy the condition 'dMX -dMY_^ 'dx 'dy " ' The factor M, which converts the equation (2) into an exact differential equation, is called, after its discoverer Euler, an Euler's integrating factor of the differential equation (2). Example. In the equation (x — yx^)dy + (j/ + xy^)dx = 0, the condition (3) is not satisfied. Hence this equation is not exact. If the equation be multiplied, however, by x'y' it will become exact ; and the method of the preceding article gives as the general integral, X 1 log = const. y xy DIFFERENTIAL EQUATIONS OF FIRST ORDER. 67 SECTION II. A Bifferential Equation of the First Order, which is Invariant under a knoiun G^, may he integrated by a Quadrature. 53. Having seen in the last section that an exact differential equation of the first order in two variables may be integrated by a quadrature, and that the know- ledge of an integrating factor of a given differential equation, which is not exact, enables us to put the equation into an exact form, we shall show in this section what it means for a differential equation of the first order to be invariaiit under a given G^; and we shall see that such an invariant equation may be integrated by a quadrature. 54. In order that an algebraic equation in the three variables x, y, y' may be invariant under a given G-y, in the same variables, it is, by Art. 42, a necessary and sufficient condition that the expression W{w) shall be zero, either identically or by means of w = 0. It was also shown. Art. 43, that if u and v are two independent solutions of the linear partial differential equation the most general form of the invariant equation w = is ^{u,v) = ^, or u-F(j;) = 0. 55. If now y be considered the differential coefficient of y with respect to x, the equation "^(3^. 2/. 2/') = will be a differential equation of the first order ; and if 68 ORDINARY DIFFERENTIAL EQUATIONS. we consider U'f to be the once-extended G-^ corresponding to the Gj in two variables, when the expression ?7'(to) is zero, either identically or by means of the equation m = 0, the differential equation of the first order, w = 0, is said to be invariant under, or to admit of, the G-^, Also we see that to find the most general differential equation of the first order which shall be invariant under a given Gj^, Uf, it is necessary to find two independent solutions of the linear partial differential equation of the first order, that is to say, we must find two independent integral- functions of the simultaneous system dx _dy _dy' i V l' One of these integral-functions may be found from the equation dx_dy and since ^ and >; are free of y', this integral-function, which we shall call u, will not contain y'. The second integral-function, which we shall denote by v, and for finding which one method has been indicated. Chap. II., Sec. 2, must contain y'. The most general invariant differential equation will then have the form v-F(u) = 0. 56. To find the integral-function u of the preceding DIFFERENTIAL EQUATIONS OF FIRST ORDER. 69 article, it is theoretically necessary to integrate a differ- ential equation of the first order, namely, dx_dy But from the form of this equation, we know, Art. 38, that tt = const, must represent the path-curves of the Gj, Hence, if the path-curves of the G^, Uf, are known, of course u is also known; and it will be remembered, Chap. III., Examples, that the path-curves of a large number of the most important G^s in the plane can be found by integrating differential equations of the first order which are exact. Thus, in a large number of the most important cases, u can be found by a quad- rature. We propose to show now that if u has been found, v can be found by a quadrature. We have already seen, Art. 43, that every infinitesimal transfor- mation in n variables can be brought, by a proper choice of variables, to the form of a mere translation. If u be known, we shall first show that in this case can be brought to the form of a mere translation by a quadrature. Let us introduce into Uf the new variables x^, y-^ ; and demand that Uf assume the form of a translation. Thus Uf, Art. 35, becomes UJ^Ui.,)^+U(jj,)^. In order that Uf shall have the form of the translation ^ in the new variables, it is necessary to have That is to say, x^ must be a solution of the partial equation Uf=0 ; 70 ORDINARY DIFFERENTIAL EQUATIONS. and since m is a solution of this equation, being by hypothesis the integral-function of the ordinary differential equation dx dy we may assume a;, = m. Now y, must be a function of x and y, which satisfies the equation and we may assume that y-^, x, and y are connected by an equation of the general form Q,{x, y, yi) = const. By differentiating this equation with respect to x and with respect to y successively, we find 'dx 3j/] 'dx ' 3S2 312 5j^^q Multiplying the first equation by ^ and the second by -q, and adding, we obtain .3i2 3i2,3r2/.3v, ?>y,\ „ %+''3^+3^U3^+''^}='^ = or, on account of the differential equation connecting x, y, and y^, .3n 3ii , 3^^Q 'dx ^2)y 3^1 But, by Art. 18, this linear partial differential equation is equivalent to the simultaneous system dx _dy _dyi that is to say, y, may be found as a function of x and y by in- tegrating this simultaneous system in the three variables x, y, y^ But we already know one integral-function of the system, namely, Xy or ?«. Hence it is obvious that^i may be found by a quadrature ; for we only need to eliminate. Art. 23, say x out of the equation by means of u = const., when we have an ordinary differential equation between y and y^, in which the variables are separate. DIFFERENTIAL EQUATIONS OF FIRST ORDER. 71 Thus, by a quadrature, we have found the new variables which ^^ make Of take the form of a mere translation, r^. But the diflFerential equations which are invariant under this translation are easily found. The extended G', in the variables x^, y^, evidently has the form U'f=^ since, Art. 48, ,'..g| _y,||.o, where ^j, iji, ?;'„ and y\ have the usual meaning. Hence to find the invariant equations, we must find two integral functions of the simultaneous system 1 ' since ^j and ij', are zero. But x^ and y\ are evidently two independent integral-functions of this system. Hence the general invariant differential equation in the variables x„ y, will have the form i2(-^i. yi)=o. If, now, we return to our former variables, this equation must take the form of a function of u and v equated to zero, say ¥{u, v) =0. But since x^ is identical with u, y\ must be a function of v ; and we can obviously assume y\—i>. Hence when the path-curves, m= const., of a given G^ ai-e known, the most general differential equation of the first order which is invariant under the given G-^ may be found by quadratures. Practically the calculations may usually be made much shorter than indicated above, since in the most important cases the variables in the simultaneous system to be integrated, dx _dy _dy' $~ v~ v' may be separated by inspection. 57. In Art. 37 the function u, which is a solution of the linear partial differential equation was called an invariant of the G^, Uf. Similarly, the 72 ORDINARY DIFFERENTIAL EQUATIONS. function v, which we saw must always contain y', and which is a solution of is called a differential invariant of the first order of the (?i, Uf. 58. If X, y, and y' be considered the coordinates of a lineal element in the plane, the equation w{x,y,y') = Q (1) represents, Art. 44, oo^ of the qo^ lineal elements ; and to demand that the equation w = shall be invariant under the (?j, U'f, is the same as to demand that the family of x^ lineal elements shall, as a whole, be invariant under U'f. For, the analytical criterion that (1) shall be invariant, means, interpreted geometrically, that the transformed (1) shall represent the same family of oo^ lineal elements that (1) itself does. But these oo^ lineal elements envelop the oo^ integral curves of (1), con- sidering this equation as an ordinary differential equation of the first order ; and since the family of lineal elements is invariant, the family of oo^ integral curves must also be invariant under the G^, U'f. Thus, if ^{x, 1/) = const (2) represent these integral curves, since (2), which does not contain y' at all, must be invariant under the extended (tj, U'f, this equation must also be invariant under the G-^, Uf; that is, by Art. 40, a condition of the form U{^{x,y))^W{^) (3) must hold, if the differential equation (1) is invariant under U'f. Conversely, if a condition of the form (3) holds, of course the oo^ integral curves (2) are invariant— and with them, the family of oo^ lineal elements (1) — or, as DIFFERENTIAL EQUATIONS OF FIRST ORDER. 73 we may say, the differential equation of the first order (1) is invariant under U'f. If, therefore, a Gj is known, of which the integral curves of a given differential equation of the first order admit, this equation, written in the form (1), always admits of the extended 0^ Hence, we may also define an ordinary differential equation of the first order as being invariant under a given 0-^, Uf, when an integral-function $ of that equation is transformed by means of Uf into a function which is itself an integral-function of the differential equation ; that is, when a relation of the form (3) exists. 59. We shall now show that a differential equation of the first order in two variables, which is invariant under a known Gj, may be integrated by a quadrature. Let the given differential equation be n{x,y,y') = 0; (4) and suppose (4) to admit of the G^, ^'fe%-^4^4- *^> We shall, for reasons explained in Art. 60, assume that Q = is not the differential equation of the oo^ path- curves of the Gp Uf. If (4) be written in the solved form X{x,y)dy-Y{x,y)dx = 0, (6) and if its integral-function be designated by oo{x, y), by Art. 16, o) must be a solution of the linear partial differ- ential equation of the first order, X^+Y^ = (7) dx dy ^ ^ Moreover, since the family of integral curves w = const, is invariant, it follows from Arts. 40 and 58 that Uia,)-=.i^ + ,^^W{a,(x,y)), (8) 74 ORDINARY DIFFERENTIAL EQUATIONS. Now if #(&)) be a certain function of w alone, # will also be an integral-function of (6), and U{^) will depend upon $ alone. For fr(*).gf.(co).gTf(.). and ft) may be removed from the right-hand member of the last identity by means of $ = $(ft)), giving thus U{^) as a function of $ alone. Since we assumed above that the curves o) = c were not the always invariant path-curves of the Gj, Uf, the function F(m) in (8) cannot be zero ; and we may easily choose as such a function of w that ?7($) = 1. For it is only necessary to determine $ so that or ^={J^. Since $ = const, represents the same family of curves that ft) = const, does, let us suppose w so chosen from the beginning that U{o}) = 1 ; that is, let us now designate by ft) the function which we have just called $. Then we have ^|ft) 3ft)^ dx dy J TT/ \ j:9<«' , 3ft) , ^ ' ^ ox ?>y Hence 3 ft) ^x'Xn-Yi' dy~X.,-Yi' , 3ft) , , 3ft> 7 Xdy— Y Since the first member of the last equation is necessarily ,, , . , 3ft) J , 3ft) J Xdy—Ydx that IS, dw = ^--ax + ::^dy = ^^ — v^^- dx dy ^ Xrj— i^ DIFFERENTIAL EQUATIONS OF FIRST ORDER. 75 a complete diiferential, the same must be true of the second member ; that is, we have the Theorem.* If a given differential equation of the first order in two variables Xdy-Ydx = admits of a known G^, whose path-curves are not identical with the integral curves of the differential equation, then is an integrating factor of the differential equation ; and the general integral raay be found by a quadrature in the form, fXdy-Ydx J X,-Yi ='^°"^*- *This theorem was first published hy Lie in the " Verhandlungen der Gesellschaft der Wisseuschafteii zu Christiania," November, 1874. By Art. 52 the equation Xdy- Vdx=0 always possesses an integrating factor, M ; and if M be known, it follows from the developments in the text that it is only necessary to choose ^ or t; in in such manner that Xr,-r$ when the given differential equation will be invariant under Uf. Although it follows from this that every differential equation of the first order is invariant under an unlimited number of O-^'a, when we speak of an invariant differential equation in this book, we shall always mean one which is invariant under a known G-^. 76 ORDINARY DIFFERENTIAL EQUATIONS. Example 1. The differential equation admits of the G-^, Uf = x^. For the extended transformation ia found by forming, Art. 48, ,_dTi ,d^ ^^^dx'ydx' which in this case, since 7; = a;, ^ = 0, is 1. Hence oy Oy By Art. 55, the expression J7'(fl) must be zero, either identically, or by means of i2 = 0. We find C/'(fi)=x|(^'-2/+.r2)+|,(^'-y+x2) = —^ + ^ = 0. Hence the condition that 12=0 shall admit of UfSs, satisfied. Now write 12=0 in the solved form, xdy — (j/ — x'')dx=0 ; since this equation admits of Xi^, the integrating factor M= has in this case the value. x.x — (y — 3^) .6~x^ and the integral is found to be, /xdy — (y — x'^)d.v — ^ — ^^^2 — = const. or, by Art. 51, y + x^ 1:=*^ =r We may at once verify that u> = const, admits of the infinitesimal transformation of the tions, Xi=x, y-i=y-\-xt. transformation of the G^, x^^- ; as well as-of the finite transforma- DIFFERENTIAL EQUATIONS OF FIRST ORDER. 77 Example 2. The diflferential equation admits of the already extended G^, For here f '(f2) has the form, U'{Q.) = ^'''^^{^y' -^-^■)^y ^{xy^ -x-f)-y'r^{xyy' ~x-f) = 2xyy'-ix+xyy'-%y'^-xyy'^^{xyy'-x-y^) = 'i.^\ t ^ = sliall admit of quation in the form, xydy — (x+y^)dx = (>; and the condition that i2 = sliall admit of C^ is satisfied. Now ■write the differential equation in the form, since it admits of ^J-'^'^-dx^y-dy y-dy" an integrating factor must be given by 1 1 Mb I'- Xr)—T^ xyy — (x+y^)2x —2x^ — xy'' Hence the integral is ( ^dy-(x+f)dx ^ or, Art. 51, 5^ = const. x' 60. The method of integration of Art. 59 fails when For this case, we see and since the first of these ratios gives the direction of the tangent to the integral curves of Q = through the point (x, y), and the second ratio gives the direction in which the point (x, y) is moved by means of the (?j, JJf, 78 ORDINARY DIFFERENTIAL EQUATIONS. the above identity states that the point {x, y) always moves on one of the integral curves of = 0. Hence the invariant family of oo' curves is none other than the family of oo^ always invariant path-curves of the G^ — each curve being separately invariant. In other words, the G-^, Uf tells us nothing new with regard to the equation Q = 0, and hence Uf is, in this case, said to be trivial with respect to that differential equation. In Art. 59 the case that Uf shall be trivial is always excluded. When Uf is trivial, since we may write i=p(x, y)X, y, = p{x,y)Y; so that Uf has the form Uf.pix,y){x%^Y^J\ ■by) Thus it is seen that every transformation of the form '(-»)(^I+''D -dy) is trivial, with respect to the ordinary differential equation Xdy-Ydx = 0. In future, we shall always disregard trivial infinitesimal transformations. SECTION III. Classes of Differential Equations of the First Order which admit of a given G^ in Two Variables. 61. Having shown that an ordinary differential equa- tion of the first order in two variables can be integrated by a quadrature when it admits of a known G-^, the next DIFFERENTIAL EQUATIONS OF FIRST ORDER. 79 step will be to find the classes of differential equations of the first order which admit of certain of the simpler (tj in two variables. From Art. 58 it is clearly immaterial whether we say that the family of oo^ integral curves of the given differential equation is invariant under a G■^, Uf, or whether we say that the differential equation itself is invariant under the Gj, Uf,OT under the equivalent once- extended Gj, U'f. 62. To finfid all differential equations of the first order which admit of a translation along the x-axis. This translation is represented, Example 1, Chapter III., by We see at once, that since ^ = 1 and 17 = 0, "^d^-ydx^^' that is, U'f= K To find the most general invariant differential equation, we must, Art. 55, find two independent integral-functions of the simultaneous system dx_dy _ dy It is evident that y and y' may be chosen as the functions designated as u and v in Art. oh ; and hence the most general differential equation of the first order which admits of a translation along the a;-axis has the form fi(2/,2/')=0; or, if solved in terms of y , i/'-F(2/) = 0. 80 ORDINARY DIFFERENTIAL EQUATIONS. In this equation the variables are separate, so that the integration may be accomplished by a quadrature. Ajaalogously, it is obvious that all differential equations of the first order which admit of the (r^ of translations along the y-axis have the form 2/'-F(a;) = 0; and are immediately integrable by quadrature. 63. To find all differential equations of the first order which admit of the G^ of affine transformations Here, since ^ = x and jy = 0, , _dri ,d^_ ''^d^~y d^-~y- Hence the extended G^ is and the simultaneous system to be integrated is dx_dy _ dy' X ~ ~ —y'' One integral-function is evidently y ; and, from x + y' -"' a second is found to be xy'. Hence the most general invariant differential equation of the first order has the form Q(xy',y) = 0; or xy' — F{y)-0. DIFFERENTIAL EQUATIONS OF FIRST ORDER. 81 Here again the variables are separate, so that the equation may at once be integrated by a quadrature. The general form of the differential equations which are invariant under the corresponding Gj of affine trans- formations along the 3/-axis, is readily seen to be y'-yY{x) = Q. In this equation also the variables may be separated by inspection. 64. To fivd all differential equations of the first order which admit of the 0^, Here, since ^ = x and >? = y, we find in the usual manner rf = 0. Hence the simultaneous system to be integrated is dx_dy _ dy' x~ y ~ ' The integral-functions of this system, usually designated by u and v, are obviously Hence, the most general differential equation of the first order which is invariant under the G^ has the form P.C. F 82 ORDINARY DIFFERENTIAL EQUATIONS. or, when solved in terms of y', This is the so-called general homogeneous equation of the first order. We may write the above equation in the form dy-Y(^dx = ^; and the method of Art. 59, gives 1 M-- !/-f(| as an integrating factor. Hence the equation written in the form dy-Y(y^dx — = is exact, and may be integrated, by the method of Art. 51, by a quadrature. Example. Given {Zx'''y + 2y^)dy + xMx = 0. This equation, being homogeneous, belongs to the class of the present article. Written in the form X? it becomes, '^y^ 'Ax^y + 'i.y^ '^'^^^ ' so that — .. =0, )dx=0, DIFFERENTIAL EQUATIONS OF FIRST ORDER. 83 or S^'y + ay^ .'fi X* + ZxY + 2y< -^ ^3i^+ 3xY + 2y* " ' must be an exact equation. It may easily be verified that such is the case ; and the general integral is found by Art. 51 to be - y — '■— = const. VX^ + T/^ 65. It should be noticed that equations of the general form (ax + by + c)dx — {a'x + h'y + c')dy = 0, (a, . . . , c' = const.) may usually be made homogeneous by a proper choice of variables. For, let the new variables be x = x — h, y = y — k, {h,k = const.) then the given equation becomes (ax + by + ah + bk + c)dx — (a'x + b'y + a'h + b'k-{- c')dy = 0. If, now, h and k are determined from the equations ah+bk +c =0, ah + b'k+c' = 0, the above equation in x and y will evidently become homogeneous, and thus may be integrated by the method of the preceding article. This method fails when a:a' = b:b'. Let us assume then a = n .a, b = n .b', {n = const.) and the original equation becomes {ax + by + c)dx—{n{ax + by)-\-c')dy = 0. Now introduce in place of y the new variable z = ax + hy; and it is readily seen that the differential equation takes the form dz , , z + c dx nz + c in which the variables may be separated by inspection. 84 ORDINARY DIFFERENTIAL EQUATIONS. Example. Given {%y-x-\)dy + {'ix-y + \)dx=0. Here the equations ah + hk+c=Q, a'h + b'k + c' — O, have the forms 2h-k + l=0, -h + 2k-l=0; so that A= — J, ^=J. Introducing the new variables the given differential equation becomes {2y-x)di/ + (2l--y)dx = 0. The general integral of this homogeneous equation is found to be x^ -xy+y^ = const. ; and if we now return to the original variables, the general integral of the given differential equation is seen to be x^ — xy+y^+x—y = const. 66. To find all differential equations of the first order which adrtiit of the 6-^ of rotations. A rotation around the origin is given, as will be remembered, by the (r^. Here ^= —y, r) = x; and hence , dri ,d^ , ,0 It is necessary, therefore, to find two integral -functions of the simultaneous system dx _dy dy' -y~ X ~r+y2" From the first equation, which may be written, xdx + ydy = 0, DIFFERENTIAL EQUATIONS OF FIRST ORDER. 85 we see that one integral function is By the method of Art. 23, we now write x^+y^ = c^, (c^ = const.) whence x = s/c'^ — y^, and ^^^=.-^^ = 0. Jc'-y2 l+y'2 The variables are separate: hence by immediate in- tegration sin"^-— tan-ii/' = fe; (?) = const.) or, sin'^ ,-^ — tan'^y' — b; But this may be written tan-'^ — tan-^2/'=6; or, taking the tangent of both sides, the second integral is found to be v = —, — ^, = const. Thus the most general invariant differential equation of the first order has the form This equation may be written — when F is put for F(xH2/^), (a;-2/F)cZ2/-(2/+a;F)dx = 0. The method of Art. 59 gives, as an integrating factor of all equations of this form, x'--\-y'' 86 ORDINARY DIFFERENTIAL EQUATIONS. so that the above equation, written in the form {x-yY)dy-{y + xF ) dx ^ ^ is exact, and may be integrated, by Art. 51, by a quad- rature. 67. To find all differential equations of the first order which admit of the Gj TT^ 3/ '^f Here t) will be found to have the value — 2y' ; so that the simultaneous system to be integrated has the form dx _dy _ dy' X —y — 2y'' Since the variables are here separate, it is seen at once that two independent integral-functions are We may write the second integral-function in the form xy.y . «2/ and since xy is itself an integral-function, we see that - must also be an integral-function. Thus we may assume xy' u = xy, v = -^; so that the most general invariant differential equation of the first order will have the form 2/'-^-F(aJ2/) = 0. DIFFERENTIAL EQUATIONS OF FIRST ORDER. 87 We may assume F = }, "^ . , and write the last equation symmetrically fi(xy) . xdy -f^{xy) .ydx = 0. Of course all equations of this form may be integrated by a quadrature ; since the method of Art. 59 gives as an integrating factor, M= ' Example. Given (x — 1/x'^) dy + (j/+xi/^)dx = 0. This equation may be written (1 - xy)xdy -\-{\ +xy)ydx = 0, so that it is seen to belong to the class of the present article. Hence an integrating factor is so that the equation ]^dy+l+^ydx=0 xnj' '' x'y is exact. The general integral is found in the usual way to be X 1 log = const. °ll xy 68. To find all differential equations of the first order luhich admit of the G^ ^.J{x)e^ ; so that the simultaneous system to be integrated has the form dx_ dy _ dy' 88 ORDINARY DIFFERENTIAL EQUATIONS. One integral-function is evidently u = x. A second may be obtained from dy _ dy' dy' or dy = {x) The most general invariant differential equation has, therefore, the form y -y-¥{x) = 0. {x) or y'-(x)+\fA(x)}dx = 0; and by Art. 59, \(x)dx e' is an integrating factor, so that fS'*''-''^ .dy-{y,i,(x) + ^{x)\e l*^^^'\d^ = DIFFERENTIAL EQUATIONS OF FIRST ORDER. 89 must be an exact equation. By Art. 51 the general integral is found in the form y = J*^^>''' j jv.(x)e- i*^^^""^ . da;+const.}. In an analogous manner it may be shown that the differential equations of the first order which are in- variant under •' dx have the general form dx-{(p{y)x+yf/-(y)dy = 0. This general equation, which, of course, may be in- tegrated by a quadrature by the usual method, is said to be linear in x, y being chosen as the independent variable. Example. Given In this linear equation the functions <^ and ^ have the form Hence e =>J\\x'; e so that ■ ((X ' «,(■ and then find two independent integral-functions of the simultaneous system dx _ dy _ d;/' Moreover, we have shown. Art. 56, that if an integral- function u, of the ordinary differential equation in two variables dx _ dy i~t' be known, the second integral-function r can always be found by a (juadrature. Practically, therefore, it is only necessary to choose i and r/ so that the ordinary equation dx d >/ will have an integrable form, either in being an exact ecjuation or in assuming one of the forms discussed, Arts. 62-69, when all differential equations of the first order whicli are invariant under the given C^and which are therert)re immediately integrable, may be found by Uf^ xf^{xy)£ + yflxy% 92 ORDINARY DIFFERENTIAL EQUATIONS. quadratures. For instance, if ^= xf-^(xy) and >] = yfzixy), the above differential equation has the form fji^xy) . xdy -f^xy) . ydx = 0, which, by Art. G7, is integrable by a quadrature. This gives us u ; and, by Art. 56, v may be found by another quadrature, so that all differential equations of the first order which are invariant under the G-^ may be found by two quadratures. This method will, in general, give rise to a new class of integrable differential equations of the first order in two variables. If desired, ^ and >; might now be so chosen that the equation dx _dy will belong to this new class. Then, of course, two quadratures will, in general, give us another new class of integrable equations, etc. 71. It will be remembered that, Art. 55, the condition that an ordinary differential equation of the first order in two variables, n{x,y,y') = Q, shall admit of a G-^, is that the expression r^vo^ .e^^j. ^"-u '^" shall be zero, either identically, or by means of fi = 0. Here, of course, U'f is put for the once-extended G^ DIFFERENTIAL EQUATIONS OF FIRST ORDER. 93 From this condition it is often possible to find the Gj of which a given differential equation of the first order admits, especially whenever the form of the equation suggests the Gy For example, the equation is homogeneous in all of its terms except one. Since, by Art. 64, ail homogeneous equations of the first order admit of the (?, we are led to suspect that the above equation will admit of a 6^1 of the form Uf= "'■'^'£. + ^y-^y («' * consts.) The corresponding extended G^ is and the condition that the given equation 12 = shall be invariant under this G^ is that V'(Q.) shall be zero identically, or by means of J2 = 0. That is, o-^'^(xo) + '(x„)ix - X,) + 4>"(x„) ^^^^^ + (4) But 4>{xi)) is yo ; 4>'{x^ is the value of y' when x = Xf,; <^"(^o) is the value of y" when x=Xf„ etc. Hence, by (2), (3), etc., y =yo +fi(xo, yo){'V - ^o) +M^'o, yo)-f^ +■■■; (5) and this is an expression for the general integral of (1). Since Xq is a particular numerical value of x, it is seen that the general integral (5) contains only one arbitrary constant, y(,. In Chapter X. we shall see that the general integral of a differ- ential equation of the m"" order may be similarly expressed by an infinite series. 73. In the following examples of differential equations of the first order to be integrated, the test for an exact difierential equation should first be applied. It will be remembered that the equation Xdy-Ydx = is exact, if dx ~ dy ' 96 ORDINARY DIFFERENTIAL EQUATIONS. and the integral may be found by a quadrature in the form I Ftia; - f(z + ^ f Yd^ dy = const. If the given differential equation is not exact, but belongs to one of the types of Arts. 62-68, it may be integrated, as already seen, by a quadrature. In case the given differential equation does not belong to one of the types established. Arts. 62-68, the method of Art. 71 should be employed to find the (tj of which the equation admits. We give below a table of types of the most important of the simpler G-^ in the plane, with the corresponding type of invariant differential equation. The reader will do well to re-establish for himself those types given below which were not established, Arts. 62-68. Group of One Parameter. Type of Invariant Differential Equation. (1) jy-2- (i)y=FCy). (2) £y^|- (2)y' = F(.r). (3)^/-^|-5|- Ci)y'=na^ + hy). It is seen that (3) includes types (1) and (2). Equations of the form {a'x + b'y + c')dy-(ax + by + c)d.x=Oma.y usually be brought to the homogeneous form. See Art. 65. (6) 6y.-y|4-..|. (6)^ = F(.H/). (7) Uf^x^^-y'^^ {'l)Uxy)xdy-flxy)ydx=0. (8) Cy^J*^">'^'-^. {S)y'-U^)y-f{x) = 0. DIFFERENTIAL EQUATIONS OF FIRST ORDER. 97 The form ^' — {'V).i/-\j/-(x)y"=0 may be reduced to this one. See Art. 69. (10) C;/.2.|+y|. (10)y>=Fg). (11) £y^^2g+^|. (11) fi(|, 2/-^')=0. (12) CA/..|+2^|. (12)^'=Fg). EXAMPLES. (1) (y2 - 4xy - 2.2:2) j;/,^ ^ (^2 _ 4^ _ 2^2^ ^^^ _ q (3) (H-,#)rfx + Jfl--Wy = 0. (4) {mdx + iidy)ain (mx + ny) = {ndx + mdy)cos{nx+my). xdx + ydy_ ydx-xdy _^ ^ ' n/1+^2+/ .r2+y2 (6) e'(a;2+y2+2.r)rf.r+2ye'rfy = 0. (7) (y-x)rfy+yrfx=0. (8) {2'Jxy-x)dy+ydx=Q. (9) xdy-{y + s/x' +y^)dx = 0. (10) (a:+y)rfy-(2/-:r)rfx = 0. (11) j;cos2rfy-(ycos"-xjrfa=0. (12) (5y + 7:r)rfy + (8y + 10^)rf^=0. (13) xdy-(j/ + -Jx^-y^)dx=(i. (14) (2j^2_^a)cj'y + (y3_2y^)rfa: = 0. (15) (.r*-2a3^3)c;y + (^y4_2^y)rfx = 0. (16) (2y-:r-l)rfy + (2:r-y4-l)rf:r = 0. (17) \ly-Zx + Z)dy + {Zy-'lx + 'l)dx = 0. (18) {xdx+ydy){x^+y^) + xdy-ydx = Q. (19) (^ + .r2)d;v+y^fl?x=0. (20) {xsnfi+f-x^)dy + {xy-yJx^+y')dx=0. 98 ORDINARF DIFFERENTIAL EQUATIONS. (21) {3flf + xy)ydx + {x'^y'^-\)xdy = 0. (22) {3?y^ + \){xdy+ydx) + {3?y^ + xy)(ydx-xdy) = Q. (23) (i/+y admits of the G^, which is not trivial, then U= Xr,-Yi is an integrating factor of (1). Suppose now that w(x, 1/) = const, represents the oo^ integral curves of (1) ; then by means of Uf each curve w = c passes over into the position of the adjoining curve w = e + Sc. At the same time every point of general position {x, y) passes through an infinitesimal distance,. Art. 29, iJi^+rfSt, of which the projections upon the, axes of coordinates are ^6t and riSt. O) THE INTEGRATING FACTOR. 101 Now draw the tangent to the curve w = c at the point {x, y) ; and lay off upon this tangent the distance 'JX'^-\-Y^, of which the projections upon the axes are X and Y respectively. The two distances >J^'^ + tfSt and -JX'^+ F"^ determine a parallelogram of which the area, by a proposition of Analytical Geometry, is {Xn-YOSt, or ^.6t. But this parallelogram, if we neglect infinitesimals of an order higher than the first, is equal in area to the rectangle constructed upon the base s/X'^-\- Y'^ with the altitude Ss, — 8s being the distance from the curve « = c to the curve w = c + Sc, measured at the point x, y. Hence we have \..St = 8s.JX^~+Y\ M St or JX^+YKSs Hence we see that if M is an integrating factor of a given differential equation (1), M is inversely pro- portional to the area of the rectangle, one side of which is the perpendicular distance, measured at a point of general position (x, y), between the integral curve through, that point and the integral curve of the family at an 102 ORDINARY DIFFERENTIAL EQUATIONS. infinitesimal distance from, that one ; while the other side of the rectangle is the distance \/X^+ Y^, rneasured off upon the tangent to the curve through the point {x, y), and from that point. 76. Let us apply the above result to a simple example. If equal distances, of length n, are laid off upon all the normals of a given curve \p-(x,y) = 0, the end points of the normals form a new curve. If, now, n varies, we find a family of qo^ curves which are called the parallel curves of the curve T/r = 0. The differential equation Xdy-Ydx = (1> may always be integrated by a quadrature, if its integral curves are a family of parallel curves. For in this case the perpendicular distance between two adjoining integral curves is constant : so that, by Art. 75, Jx^+ 72 must be an integrating factor. Hence, if it is known that the differential equation (1) represents a family of 00^ parallel curves, JX^+Y^ is an integrating factor of (1). Conversely, it is easy to see that if M= y/X^+ T' be an integrating factor of (1), the distance between two adjoining curves must be constant, and the curves are parallel. ORTHOGONAL TRAJECTORIES. 103 The 00^ involutes of a given curve form a family of parallel curves ; and hence their differential equation may always be integrated by a quadrature. For example the involutes of the parabola are represented, as is shown in the Differential Calculus, by the equation 2(x + Jx^-y)dy + da; = 0. 1 Hence M = J^{x^■Jx^-yf->r\ must be an integrating factor of the above equation, as may be at once verified. 77. An orthogonal trajectory is a curve which inter- sects at right angles each member of a given family of Qo^ curves. A family of (Xi^ curves, represented by a differential equation Xy'-Y=() (1) will evidently have oo^ orthogonal trajectories ; and their differential equation is readily obtained from (1). For, at any point of general position (x, y), the integral curve of (1) through that point is perpendicular to the orthogonal trajectory through the point ; hence, if y' be the tangential direction of the integral curve, ; must be that of the orthogonal trajectory. Thus if we substitute in (1) ; for y', we obtain the differential equation of the oo^ orthogonal trajectories of the integral curves of (1), in the form X+Yy'=^ (2) Reciprocally, the integral curves of (1) are the or- thogonal trajectories of (2). 104 ORDINARY DIFFERENTIAL EQUATIONS. Example. It is required to find the orthogonal trajectories of the hyperbolas xy = a. (a = parameter) The differential equation of these oo^ curves is obviously xdy+ydx=0, or xy'+y = Q. Writing — -, in place of y, we find as the differential equation of the orthogonal trajectories x-yy' = 0, or xdx-ydy=0. The variables are here separate, so that we find at once the integral curves which is also a family of hyperbolas, x'^-y'^=c\ (c= parameter) 78. A family of oo^ curves in the plane is said to be isotherTnal when, together with their orthogonal trajec- tories, they form a network of infinitesimal squares. If Xdy-Ydx = •(1) represent a family of isothermal curves, their orthogonal trajectories will. Art. 77, of course, be represented by Xdx+Ydy = 0, •(2) ISOTHERMAL SYSTEMS. 105 and both of the equations (1) and (2) may be integrated by quadratures by means of the geometrical interpre- tation of the integrating factor, Art. 75. For, if we consider any two adjoining curves of each of the families (1) and (2) which form a small square at the point {x, y), the breadth, Ss, of the strips enclosed by both pairs of curves is the same, since Ss is the side of the infinitesimal square. Hence M = —-- ^- - is an integrating factor of both (2) and (1). But if two ordinary differential equations of the forms (2) and (1) have a common integrating factor, this factor may be determined by a quadrature. For, if M be the common integrating factor, by Art. 52, M must satisfy the equations ■dMX dMY_ ■dx ■*■ dy a¥F_3MZ_ ■dx dy ~ ' or j-d]ogM d}ogM^_dX_dY dx ^ dy dx dy' ydhgM ,. 3 log if _ dYdX ,„- ^ dx ~^~^^-"^+3^ ^^^ From the last two equations — :— — and —^ — may ^ dx dy '' be determined as functions of x and y ; and, if these quantities satisfy the condition of integrability, Art. 51, d 3logilf _ d alogilf dy dx ~dx dy ' ' we may find log M, or M itself, by a quadrature from the exact equation 106 ORDINARY DIFFERENTIAL EQUATIONS. If the above condition of integrability were not satisfied, (1) and (2) would have no common integrating factor ; and hence the families of integral curves would not be isothermal. Thus, that (3) shall give such values for — ^ — and — ^ — as satisfy (4) is a necessary condition that (1) shall represent an isothermal family. This condition is also suiEcient; for if (4) is satisfied, (1) and (2) have a common integrating factor — that is, the quantity designated as Ss above must be the same for both families of integral curves, and these integral curves form a net-work of small squares, or are iso- thermal. If (1) represents an isothermal family of curves, there- fore, the common integrating factor M, of (1) and (2), may be found by a quadrature; and the equations (1) and (2) may be integrated by another quadrature each. Example 1. The differential equation 'ixydy-(j/''-x'')dx=0, (5> represents an isothermal family of curves. For the orthogonal trajectories are represented by (y'^-x')dy+^dx = 0; (6) so that the equations (3) have the forms TT 31ogi/' -ix "dlo^M _ -iii_ ?)x ~x'+9/^' 3y "x^+y"' and it may be immediately verified that the condition (4) is satisfied. Thus (5) represents an isothermal family ; and the integrating factor of (5) and (6) is obtained from rflogifs -2{^,rf^ + ^prfy} =0 ISOTHERMAL SYSTEMS. 1()T by a quadrature. We find \ogM=-'2.\og{x'^+f); and hence M=-r^ jr^. It may be at once verified that this is an integrating factor of (5) and also of (6). Hence, from (5) and (6) respectively, we find by quadratures x y - const., --,- — 5 = const. as integrals. The first isothermal family is that of all circles which touch the ?/-axis at the origin ; the second is that of all circles which touch the ^-axis at the origin. Of course equations (5) and (6) might also have been integrated by the method of Art. 64. 79. In the following examples, such differential equa- tions as represent curve-families consisting either of isothermal or of parallel curves may be integrated by the method of Arts. 76-78. It may usually be seen, from the geometrical meaning of the equation given, whether the orthogonal family will be isothermal or not. Those differential equations which represent orthogonal trajectories which are neither parallel nor isothermal curves may be integrated by Art. 73. EXAMPLES. Find the orthogonal trajectories of the following curve-families : (1) The straight lines y = ax. (a = parameter.) (2) The parabolas y'^ = iax. (3) The circles x^+y'^ = a'^. (4) The parabolas y- = ia{x + a). (5) The ellipses ^2+f2=l- (a = const.., 5 = parameter.) (6) The circles 3(^-^x+y^-^ay + a^=Q. (7) The ellipses ^^%^'^^' (w = parameter.) (8) The circles x'+y^ + ax-l=0 108 ORDINARY DIFFERENTIAL EQUATIONS. (9) The confocal conies -^2 — •)+f2 = ^- (6 = parameter.) (10) The parallel curves y — a=tf)(x), which result from translating the curve y = <^(x) parallel to itself along the y-axis. (11) Apply the result of Ex. (10) to the case of the semi-cubical parabola (12) Showthat the differential equation of the orthogonal trajectories of the curves in polar coordinates F(p, e, c)=o is obtained by eliminating c between the above equation and dd de f^dp'^- (13) Find the orthogonal trajectories of the curves p = log t^n 6 + a. (14) Find the orthogonal trajectories of the curves which result from rotating the curve around a fixed point iu the plane. (15) Apply the result of Ex. (11) to find the orthogonal trajectories of the circles which result from rotating the circle p = b cos 6. around one end of its diameter. CHAPTEE VI. DIFFERENTIAL EQUATIONS OF THE FIRST ORDER, BUT NOT OF THE FIRST DEGREE. SINGULAR SOLUTIONS. 80. We propose to give in Sec. I. of this Chapter methods for integfating some of the simpler differential equations in two variables which are of the first order, but not of the first degree. In Sec. II. we shall see that a differential equation of a degree higher than the first is sometimes satisfied by a function which is not a function of the "integral- function" of the equation. This peculiar function, equated to zero, constitutes what is known as a "Singular Solution" of the given equation. A simple method for finding the Singular Solution — when one exists — of an invariant differential equation of the first order will be given. SECTION I. Differential Equations of a Degree Higher than the First. 81. In Art. 55 it was shown that the condition that the differential equation of the first order n{x,y,y') = ii 110 ORDINARY DIFFERENTIAL EQUATIONS. shall admit of a given G^ is that the expression U'{Q) shall be zero, either identi- cally, or by means of fi = 0. It is clear that this condition is independent of the form of n, and hence of the degree of the equation Q — 0. It will be borne in mind, however, that in order to apply the method of Art. 59 to integrate the equation fi = by a quadrature, when it is known that this equation admits of a given G^, U'f, it is necessary to solve fi = in terms of y', in the form, Xy'-Y=0. The algebraic solution of Q = in terms of y' is not always simple ; and in Cases II. and III. below we shall indicate methods by means of which that work may sometimes be simplified or avoided. 82. Case I. Suppose that the equation Q{x,y,y') = Q (1) may be solved, algebraically, in terms of y' in such manner that the resulting roots will be rational functions of X and y. Thus, if (1) is of the %'*' degree, it may be written iy'- y))iy'-n{x, y))=o, ...(2) where the (p^, ...,n are rational functions of x and y. In this case, since (1) can be resolved into the linear factors in (2), (1) is called a "decomposable" equation. But (2) is satisfied by writing y' - 4>x{^> y) =0, y'- lx, y)=o,...y'- (pn{x, y)=0;...{S) and if the general integrals of the n equations (3) have been found, by the methods of Chap. IV., in the form y-^^{x,y,c^) = 0, y-^lx,y,Ci) = 0,...y-^J^x,y,Cn) = 0, DIFFERENTIAL EQUATIONS OF A HIGHER DEGREE. HI the general integral of (1) will have the form {y-*i(a;,3/,Ci)}{2/-*2(a;,i/,C2)},...{i/-*„(a3,2/,c„)}=0.(4) But equation (4) loses nothing of its generality if we assume the arbitrary constants all equal to one constant, say c. For, in order to find any value of i/, it is necessary to equate to zero one of the n factors on the left-hand side of (4), which gives an equation of the form 2/-*t(a;, y, c) = (5) Now since c is an arbitrary constant, by giving c all possible values, the form (5) may be made to contain all the integrals which may be derived from the corre- sponding A:"" factor of (2). Hence we find as the general integral of the original decomposable differential equation (1): {2/-*i(x, y,c)) ... {y-^n{x,y,c))=0. Example. Suppose (2) to be of the second degree, of the form y'^-{x+y)y' + xy = (6) This equation may be written 0/'-^)C/-y)=o, ■whence «/'-x=0, y'-.y=0. Integrating the last two equations we find y--2-=0, y-ee'=0. so that the general integral of (6) is 83. Case II. If the given equation n{x,y,y') = (} can be readily solved with respect to y, in the form y = (2/.3/') = 0, (8) it is sometimes best to differentiate (8), regarding y' as variable as well as x and y, and substituting ^ for dx. A differential equation of the first degree in terms of y and y' must result. Integrate this equation by the methods of Chapter IV., and eliminate y' between the resulting equation and Q = 0. SINGULAR SOLUTIONS. 113 Example. Given yy"^->r%xy'=y. Hence x=v ~^ , ■=' 2y' and dx=^^dy-y^-^dy'. Putting for dx, -^, we find at once, ^ + ^3^ = y y' The general integral of this equation is yy' = c. Eliminating y' from the first equation, the general integral required is found to be 85. An important equation of the form which is known as Clairaut's equation, and which may be integrated in a manner analogous to that employed in Case II., will be treated separately in the next Chapter. SECTION II. Singular Solutions. 86. It will be remembered that in Art. 59 it was shown that if a given differential equation of the first order i}(x,y,y') = 0, (1) admits of a known (r^ the differential equation may always be integrated by a quadrature, ^?'0V2(ied that the infinitesimal transforma- tion Uf is not trivial with regard to fi = : that is, provided that the path-curves of Uf do not coincide with the integral curves of fi = 0. 114 ORDINARY DIFFERENTIAL EQUATIONS. But now the question arises, may not a limited number of path-curves of the G'l coincide with particular integral curves of fl = 0, when the G-y is not trivial ? It will be found that such may be the case. For, along curves which are at once path-curves of the G^ and integral curves of fi = 0, the value of y' given by the G^ must coincide with the value of y' given by fl = 0. Hence, to find such curves we only need to substitute in n = 0, and the resulting equation ,.(2) will give the path-curves of the G^ which are also integral curves of (1), if such exist. But it is easy to see that we may also find in this way the equation to a curve which is a path-curve of the Gj and which satisfies the given differential equation (1), but which is not a particular integral-curve of (1). SINGULAR SOLUTIONS. 115 For it may happen that the family of integral-curves (1) have an envelope, and if so, it is clear that the equation to the envelope will satisfy the differential equation (1) ; for at any point on the envelope the direction of the tangent to the envelope is the same as that of the tangent to either of the two consecutive curves of the family, which, from the Differential Calculus, we know must coincide in that point. Hence the value of y at any point on the envelope will satisfy the differential equation ; and the equation to the envelope is called a "Singular Solution" of (1). Since at any point on the envelope two values of y' given by (1) must coincide, it is clear that equation (1) must be of at least the second degree in y' in order that the integral-curves may have an envelope. But now the family of integral-curves of (1) is in- variant under the transformation Uf; hence it is clear that the envelope of the family, if one exists, is an invariant curve of which the points are interchanged by means of the transformation Uf. In other words, the envelope must be a path-curve of the (?j, Uf, of which (1) admits. To find this particular path-curve, we only need to substitute -| for y' in (1); and the resulting curve, or curves, must, as we saw, be those curves in the plane for which the values of y' given by the differential equation (1), and by the G-^, are the same. Hence, we find by this method the singular solution of (1), if one exists; and, occasionally, as indicated above, a limited number of particular integral-curves of the differential equation, which are, at the same time, path-curves of the The particular integral-curves may be distmguished from the singular solution by the fact that the equation to a particular integral-curve may always be obtained from the general integral of (1) by assigning a special value to the constant of integration, while the equation to the singular solution cannot be so obtained. 116 ORDINARY DIFFERENTIAL EQUATIONS. If the equation (2) breaks up into factors, each factor must be separately examined to see whether it is a particular integral or a singular solution. It may be remarked that ^ and r) cannot both be zero along the enveloping curve of an invariant family ; for, as we saw above, the points of the envelope are inter- changed when the curves of the invariant family are interchanged, whereas all points on curves along which ^= ri = Q, are absolutely invariant. Example 1. Given ^'^y^cos^a — iy'xy sin^a +y'^ — .r^in^a = 0. This equation is homogeneous, and hence is invariant under the G^, Thus, according to the above theory, we find the singular solution, if one exists, by substituting in the above equation - for y'. We obtain, after an obvious reduction, the two equations x^ = {x^ +y^) eos^a. The general integral is found by Art. 59 to be x'^+y'^-2cx+c'cos!^a = 0; (c=const.> and hence we see that 3^2 + ^2 = 0, which may be obtained from the general integral by assigning to the arbitrary constant c the value zero, is a particular integral ; while ^2 = (^2 + y 2) cos^a, which may be written y= ±a;tana must constitute a singular solution, since these equations satisfy the given differential equation, and cannot be obtained by assigning any special value to c in the general integral. SINGULAR SOLUTIONS. 117 Example 2. We know, Art. 71, that the differential equation y - ixyy' + 83/^ = admits of the G-^, To find the singular solution, if one exists, we substitute for y', in the above equation, -i. Hence X ^■ly^-4^y'^ = 0, or y = 0, y- — = 0. The general integral is y = Ci{x-cf; and hence we see that y = is a particular integral, while is the singular solution. ^-27=° Example 3. The differential equation y'%l-x')-x^ = 0, being free of y, admits of the 6^1 Since for this G^, By = l, Sx = 0, we have Sx " !/ Therefore we write the above equation (1_^2)_^ 0; and, substituting for y' the value co , we find x=±\. This is a singular solution, since the differential equation possesses the general integral x^^-(y-af='^- The geometrical meaning of the singular solution in connection with the 00 1 curves represented by the general integral is obvious. 118 ORDINARY DIFFERENTIAL EQUATIONS. EXAMPLES. Integrate the following diflFerential equations, finding the singu- lar solutions, when such exist, as well as the general integrals. For types of invariant equations, see Art. 73. (l) y^-by' + Q = 0. (2) y2_„y = o. (3) x^-^ + -ixyy' + ^y^=Q. (4) y'{y' +y)=i:{x+y). (5) y^ + 2x/2 - yY^ - ^-yh/ = 0. (6) y'^ + 2yy'cot x =y\ (7) .v=(l +x)f'. (8) yy'^+2xy- -y = 0. (9) Zxy^-Qyy' + x+'2.y^0. (10) Zyh/''-'ixyy' + iy'^-x'=0. (11) y=-xy' + x*y'K (12) xy'^-iyy' +ax = 0. (13) y = ay' + hy'\ (Art. 83.) (14) x'^+y=y'K (15) /=.r2(l +/2). (16) y=y'^ + '2.y'\ (17) ■ifhi-^'ianf=y. (Art. 84.) (18) a:=y+logy. (19) ;f^'*=l-|-y2. {W)my-rucy'==yy'\ (21) ^2(y'2+2) = 2y3/3 + .r3. (22) yy'i + {x-y)^=^x. (23) ^^^'2 - 2yy + 4a: = 0. (24) /* = 4y (ay - 2y)2. (25) 4y'2ar (a; - a)(a: - 6) = {3j-2 - 2.r(a + 6) + a6p. (26) ah)y'^ - ixy' +y = 0. (27) y'^ + 2a^y = 4^=^. CHAPTER VII. RICCATI'S EQUATION AND CLAIRAUT'S EQUATION. 87. We propose in this Chapter to make brief mention of two important historical differential equations of the first order, which are known respectively as Riceati's equation and Clairaut's equation. The treatment of these equations sketched here will be the same as that of the ordinary text-books : for, although both equations may be treated most advantageously from the standpoint of the Theory of Transformation Groups, that method would require a more extensive knowledge of these groups than it is advisable to give in an elementary text-book. SECTION I. Riceati's Equation. 88. This equation takes its name from that of an Italian mathematician, Riccati, who was the first to discuss it. The general form of Riceati's equation is ^-^{x).y^--ir{x).y-x{x) = (i; (1) but this equation can only be integrated in a few special cases : and the particular form usually discussed is a!^-a2/+V = ca!", (2) 120 ORDINARY DIFFERENTIAL EQUATIONS. where a, h, c, n, are certain constants. By introducing into (2) the new variables 2 = a;», w = ^, that equation takes the form -T-^ — u = z" (.o) dz a a a special form of (1), which is itself sometimes designated as Riccati's equation, instead of the more general equa- tion (1). The equation (2) happens to be much more easy to discuss than equation (3) ; and it is easy to deduce from the condition that (2) shall be integrable the condition that (3) vshall also be integrable. We shall first show that equation (2) is always integrable when n = 2a; then we shall show that the integration of (2) may always be made to depend upon this case when — ^ — is a positive integer. 89. Case I. The equation x^-ay + by^ = cx'' (2) is always integrable when n = 2a. Let us assume y=x''v ; then (2) becomes x'l-a- — \-bv^ = cx"-'^; dx and if 71 = 2a this equation becomes 1 „dv ^ „ x^'"--, — \-bv^ = c, dx dv _ dx_ ^^ c-6ti2~a;i-'' ^'^^ In this equation the variables are separate, so that it may be integrated by a quadrature. If we return to Rice ATP S EQUATION. 121 the original variables, we find the exact differential equation by^-cx^^ +^"-'^^-0 (4) of which the general integral is given by * .<^e « +1 ^ \bJ 2(6c)V Ce » -1 y = (-g\^Han{c-^-y^''}, according as b and c have the same or opposite signs — C being the arbitrary constant of integration. 90. Case II. The equation x-£^-ay + hy^ = cx^ ^2) is always integrable ivhen —^ — is a positive integer. Let us assume ■y=-4H — . where 4 is a constant to be determined. The equation (2) is easily seen to take the form -aA + bA^+{n-a+2bA)- + b—.-'^.-^ = cx^....{5) We shall choose A so that the constant in this equation shall be zero; thus we may choose A=r, or .4 = 0, so that there are two subdivisions for this case of the problem. 122 ORDINARY DIFFERENTIAL EQUATIONS. (1) If, in the first place, -4 =t, the last equation, after a slight reduction, takes the form, x'^^-{a+n)y,+cy^^ = hx- (6) It is seen that (6) is of the same form as (2), except that h and c have changed places, and a has been changed to a + n: and this change was brought about by substituting in (2) a , x" in place of y. Hence, if in (6) we make the substitution a + n , X" it is clear that (6) will take the form x'^^-{a + 2n)y^ + hy^^ = cx-, (7) where 6 and c have again changed places, and a-\-n has become a + 2n. Thus, if X successive substitutions of the above forms are made in the equation (2), that equation will take either the form x^-{a+\n)yy^ + cy^^ = hx^, (8) *^~^'^+^''^2/x + ^3/x' = cx», (9) or the form according as \ is odd or even. But by Case I., the equations (8) and (9) are in- tegrable if n = 2{a, + \n). Rice ATP S EQUATION. 123 that is, if !ir_!^=x. (2) Secondly, let us assume A=0. Then (2), by means of the substitution a;" is readily seen to take the form ^2x-in-a)y, + cy^^ = bx«, (10) an equation which is identical with (6) except that a has become — a. If the preceding series of substitutions are now made, the final result will be found to be the same except for the sign of a. Thus the equation (2) will be integrable when 71+ 2a Combining these results we see that the equation is integrable whenever —^ — is a positive integer. From the nature of the substitutions employed in the above two cases, it is clear that the general integral of Riccati's equation, when that equation is integrable by the above method, will be given in the form of a finite continued fraction, the last denominator of which is to be found by a quadrature. 91. We found, in the last article, as a condition of 71 ± 2a integrability, that —z should be a positive integer, say X. ^™ 124 ORDINARY DIFFERENTIAL EQUATIONS. If we assume — r^ =X, from (1) Art. 90, we have the series of substitutions a , a;" _a + 2n x^ a + (\-l)n , a;" where fi has the value b or c, according as X is odd or even. From these equations we have a a:" ^"~6 a + n , x^ « "^+..., (11) D the last denominator of the finite continued fraction being a + (\-l)n x" The value of y\ is to be determined by a quadrature from one of the equations (8) or (9). These equations may now be put into exact forms, analogous to (4), writing a + Xn for a : x''^^-dy^-(a+Xn)y,a^^^'^-^dx ^^^^^_, cy-f^ — bx"' and JJi — \^ — 'JiA hx«+''"-icZx = 0, ...(13) by-^—cx^ X being supposed odd in (12) and even in (13). Rice ATI'S EQUATION. 125 If now we assume 71 + 2a it is easy to see that y will have the value _ cc" "~n — a a;" r 2n — a ?^-V..., (u) where the last denominator is (\-l)n-a a;" Also, 1/x is to be found from one of the following exact equations, which result from (12) and (13) by changing a into — a : '^ — ^^ — 5 — —^ hx'^""""^(^a; = 0, ...(15) '^y\ ~ "^ ^ — ^ — = ^-^ ha;''"-'"-ida; = 0, ...(16) oy\—cx^ X being odd in (15) and even in (16). Thus, when — ^ is a positive integer, (11) in con- Jilt nection with either (12) or (13), according as \ is odd or even, will represent the required general integral. When —^ — is a positive integer, (14) in connection Act with either (15) or (16), according as X is odd or even, wUl represent the required general integral. 92. By making use of new constants, the equation (3) which is itself sometimes designated as Riccati's equation, may obviously be written ^+hu^ = cz^ (17) dz 126 ORDINARY DIFFERENTIAL EQUATIONS. If, in place of z and it we introduce the new inde- pendent variables x, and y = ux, (17) becomes x^-y + by^ = cx'^+' (18) But we know that (18) is integrable when (m + 2)±2 _ 2(m + 2) ~ ' where \ is a positive integer. This is therefore the condition that the special form (17), of Riccati's equation, shall be integrable. Making use of the negative and of the positive signs in succession in the above condition, we find — 4X ™ = 2X3T' (19) -and m=- ^^^~l ^ (20) By changing, in (20), the integer X into X+l, which is obviously allowable on condition that X may assume the value zero, as well as any positive integral value, {20) may be written — 4X ™ = 2XfI- (21) Here, if X = 0, m = 0; and since for X = in (19) we also have m = 0, it is clear that X may admit of the same series of values in (19) as in (21). Combining these results, we see that Riccati's equation, in the special form (17), is integrable whenever -4X "^ = 2-xTI' X being zero, or some positive integer. When the negative sign is used in the last equation, the general integral is given by (11) in connection with <12) or (13) according as X is odd or even. When the RICCATrS EQUATION. 127 positive sign is used, the general integral is given by (14) in connection with (15) or (16) according as \ is odd or even. Example. Given the equation du „ „ -8 dx"^^^ (22) This is a case of equation (17). By substituting ylx for u, the equation takes the form of (18), ^%-y-y'=^x~"^ (23) The condition of iutegrability (19) '« = 2X^' (19) gives _? = -i:iA, ^ 3 2A-1' or A, = 2. Hence, the integral of (23) is given by the equation (11) in connection with (13). Here we have a=\, A.=2, n— —% a+n.A=-J, 6=-l, c=2; hence (13) becomes This is an equation of the form (4), where the a, 6, and c of that equation have the respective values -i, -1, 2. Thus, since h and c have opposite signs, the integral of the last equation is y^ = ^'2.r "*tan( C+ 3^2:;;"*). This value of yj substituted in y = l+^+f^ ^"'^ gives the general integral of (23) ; and if in that result we restore to y its original value ux, we find the general integral of (22). 128 ORDINARY DIFFERENTIAL EQUATIONS. SECTION II. Glair aut's Equation. 93. The equation of the form y = xy' + (t>{y'), (1) is generally known as Clairaut's equation. Although an equation of the first order, it is not usually of the first degree. Difierentiating (1), regarding y' as a variable, as well as X and y, we find y' = y'+{x + ^\y')}^; (2) from which follows either ^' = 0. dx or x+^\y') = (3) From the former of these equations follows y' = c, (c = const.) so that, from (1), the general integral must have the form y = cx + {z). Differentiating, we find ydx+xdy--^dz=0. In order that this equation may be equivalent to the first one, we must have d_ xy dz 6 — 0* or ^^_i_. dz z — b TOTAL EQUATION IN THREE VARIABLES. 137 Here the variables are separate, so that an immediate integration gives (^ (2) = c (z - h). (c = const.) The required general integral is, therefore, xy=c{z — h). 100. We shall give a method, based upon geometrical considerations, by means of which the total equation (1) Pdx-\-Qdy+Rdz = Q (1) may, when the condition (5) of the last section is satisfied, be integrated by integrating one ordinary differential equation of the first order in two variables. Since (5) is satisfied, (1) has a general integral of the form ^{x,y,z) = c; (2) and (2) represents ^ surfaces in space, called the integral surfaces of (1). If we cut these surfaces by a family of oo^ planes, say, z=x+ay; (a = const.) (3) then for each value of a we obtain 00 * curves of inter- section of one of the planes with the oo* surfaces (2) ; and these curves are represented by a differential equa- tion in X and y. To find this differential equation, we only need to eliminate z and dz from (1) by means of (3) and of dz=dx+ady ; giving the differential equation in the foim {x, y, a)dx+\lr{x, y, a)dy = (4) If, now, (4) has been integrated, by the methods of Chap. IV. — a being an arbitrary constant — we may easily find the 00^ surfaces (2), since we know their 00* curves of intersection with the planes (3). For the 00^ curves of intersection which pass through one point on the axis of the family of planes (3) will in general form one of the integral surfaces (2). 138 ORDINARY DIFFERENTIAL EQUATIONS, Now a point on the axis of the planes (2) is evidently determined by 2/ = 0, x — k; ((c = con8t.) and if the general integral of (4) be W(x, y, a) = const., (5) in order for the curves (5) to pass through the point 2/ = 0, x = K we must have W{x,y,a)=W{_K,0,a) (6) When a varies, (6) represents the oo^ curves through the point y = 0, x = k; and if k also varies we obtain successively the co^ curves through each point on the axis of (3). That is, if by means of (3) we eliminate a from (6), we obtain the integral surfaces required in the form T(.,!,,i=5)-F(.«.^) = 0. Thus the complete integration of (1) has been accom- plished by integrating one ordinary differential equation, (4), in two variables. If the constant a happens to factor out of (4), some other family of planes must be used in place of (3). This method is theoretically better than that of Art. 99, since only one differential equation in two variables has to be integrated ; but the differential equation (4) is often more diflScult to integrate than are equations (6) and (9) of Art. 99. Example. Given {y + z)dx+dy + dz = 0. This equation evidently satisfies (5), Sec. I. ; and if we write z=x+ay, the equation (4) becomes TOTAL EQUATION IN THREE VARIABLES. 139 This is a linear equation, with the general integral W{x, y, a) = «*(y + Y^j = "' (fl=const.) Hence equation (6) has the form or, writing for a, ^_y'+yz _^ K.y y+z—x y+z—x that is, e'(2/ + z) = const. EXAMPLES. Integrate the following ordinary differential equations in three variables, after verifying that the condition (5), Sec. I., is satisfied: (1) (y+z)dx + (z+x)dy+{x+y)dz=0. (2) xzdx + zydy=(y^+x^)dz. (3) (x-3y-z)dx+(2y-3x)dy + (z-x)dz=0. (4) ayh^dx + bz^sfidy + cx'y^dz = 0. (5) (y+afdx+zdy-(y+a)dz=0. (6) {y^+i/z)dx+{xz+z^)dy+{y^-xy)dz=0. (7) {2i^+2xy+2xz^+\)dx+dy+2zdi=0. CHAPTER IX. ORDINARY DIFFERENTIAL EQUATIONS OF THE SECOND ORDER IN TWO VARIABLES. 101. In this chapter we propose to develop a theory of integration for ordinary differential equations of the second order in two variables analogous to that developed in Chapter IV. for ordinary differential equations of the first order. The liTiear differential equation of the second and higher orders will be treated separately in Chapter XL SECTION L Exact Differential Equations of the Second Order. 102. If an ordinary differential equation of the second order of the form ^{x,y,y',y") = Q be given, we know that the complete primitive, or general integral, is an equation involving two inde- pendent arbitrary constants, Cj, c^, of the form ^i(a;, 2/, Ci, C2) = 0. It may be shown by means of the Theory of Functions that if the complete primitive of Q = be written in the form y-W{x,c^,c^ = 0; EXACT EQUATIONS OF SECOND ORDER. 141 and if be a second equation, which, when treated as a complete primitive, gives rise to the same differential equation of the second order, = 0, then it must always be possible to choose the a^, a^ as such functions of c^, c„ say that W(x, Ci, C2) = w{x, \, \). If the above complete primitive 'SP'i = be differentiated, an equation of the form will result, from which, by means of "^-^ = 0, c^ and c^ may be successively eliminated, giving rise to two inde- pendent differential equations of the first order of the form *i(a3, y, y', Ci) = 0, *2(a:, y, y\ c^ = 2/" with respect to y'. If the expression thxis found be differentiated totally with respect to x, and the result be subtracted from the first member of (3), the remainder must be a differential expression of an order not higher than the first. Also since this remainder is the difference 144 ORDINARY DIFFERENTIAL EQUATIONS. of two exact differentials, it must itself be an exact diflferential, and y' can occur in it only to the first degree. Its integral, together with the terms already found by integrating with respect to yf, will be the integral of the whole equation (3). Example. The equation 3cyf-\-an/^-y]/=0, (4) is exact. For the term involving y is xyy", and this being integrated with respect to y' gives xyy'. The last expression when differ- entiated totally with respect to x gives xyf+x}^'^-\ry^. Subtract this result from the first member of (4), and the remainder, - 'i.yy', will also be exact, having for its integral -y^. Hence equa- tion (4) is exact, and a first integral is If now we divide (4) by x^, it will be seen that a second first integral is Hence the complete primitive is SECTION II. Lie's Differential Equations of the Second Order. 104. Of course not every difierential equation of the second order is exact; and there is no general method for integrating all such equations when not exact. It will be our object, however, to show in this paragraph, how the knowledge that the given difierential equation of the second order admits of, or is invariant under, a given G-i can be used to reduce the problem of integration in a number of the most important classes of difierential equations of the second order. These invariant differ- ential equations of the second order are sometimes called " Lie's Equations of the Second Order." LIE'S EQUATIONS OF SECOND ORDER. 145 105. Suppose that the infinitesimal transformation of a given 0^ in two variables be twice extended by Art. 47 ; in the four variables x, y, y', y" the twice- extended infinitesimal transformation will have the form where „'^^-v'^ J'^^-v"'^ ' dx " dx' ' dx " dx The necessary and sufficient condition that an equation in the four variables x, y, y', y", of the form Q(x,y,y',y") = 0, may be invariant under the Gj U"f is, by Art. 42, that the expression U"(Q) shall be zero, either identically, or by means of fj = 0. It was also shown in Art. 43 that if u, V, and w be three independent solutions of the linear partial differential equation ^•^=^^ + "3^ + "^ + '' 37' = ^' the most general form of the invariant equation, Q = 0, is obtained when fi is expressed as an arbitrary function of u, V, and w, say Q.(x,y,y',y")^Y(u,v,w) = ^. Or, if we choose, we may solve F = in terms of one of the three quantities u, v, or w, say in terms of w, and thus put the most general invariant equation in the form w — '^{u, v) = ^. But X and y may be interpreted as point coordinates in a plane ; and y' and y" as the differentials dy d^y dx' dx^' 146 ORDINARY DIFFERENTIAL EQUATIONS. respectively. In this case the equation Q = is a differ- ential equation of the second order in the variables x and y ; and if the expression U"{iT) is zero, either identically or by means of D = 0, the differential equation f2 = is said to be invariant under, or to admit of, the Q.^ U"f. Now the differential equation Q = represents a doubly infinite system of curves in the plane; and that the equation n = shall be invariant under U"f means that the system of curves must be invariant under U"f also. For, if we designated the new variables, as usual, by *i' Vv Vi' 2/i") S'lid the transformed equation by fli = 0, then, by hjrpothesis, Q^ must have the same form in the new variables that fi had in the variables x, y, y , y" : that is, 12^ = must represent the same family of curves that Q = represented. But this family of curves will also be transformed by ^•^-^30! + ''32/' and at any point of general position (x, y), y' and y" will receive the same increments, Art. 47, by means of Iff, that they receive at that point by means of U"f. Hence, a point P, in the plane which satisfies the system of values X, y, y', y", will always be transformed by means of Uf to the point P^ which satisfies the system of transformed values Xj, y-^, y( y(' ; and it is clear that since P passes to the same position Pj under the trans- formation Uf that it does under TJ"f, and since y' and y" are transformed by Uf exactly as they are by U"f, the family of curves, represented by f2 = 0, must also be invariant under Uf. Thus it is clear that the condition that a differential equation of the second order Q{x,y,y',y") = ^ shall be invariant under the twice-extended G.^ U"f, is the same as that the family of oo" integral-curves of = shall be invariant under Uf LIE'S EQUATIONS OF SECOND ORDER, 147 When the condition that TJ"{D,) is zero, either identically or by means of Q = 0, is satisfied, we sometimes say, for brevity, that Q = is invariant under the 0-^ Uf, instead of under the twice-extended Gj U"f. 106. We have a general method. Art. 23, for deter- mining the quantities u, v and w of the preceding article, and we have seen. Art. 56, that when u has been found from the differential equation dx_d^ i~ 1 ^^^ then V can be found as the second integral-function of the system i 1 n by means of a quadrature. Unless the path-curves of the Gj, which are represented by u = const, are known, it will be necessary to perform an integration to find u from the above differential equation of the first order (1); but we shall show that when u and v have been found, w can be found by mere processes of differ- entiation. Also w, as will be seen, must necessarily contain y" ; and as v(x, y, y') was called, Art. 37, a differential invariant, of the given 0-^, of the first order, so wix, y, y , y"), fo^^ reasons which are obvious, is called a differential invariant of the given 0^, of the second order. 107. We shall now show that when u and v have the meaning of the last article assigned to them, and are known, that w, the third independent integral-function of the system dx_dy_ d^ _ dy" ,„-. i~ 1 v v" can be found by differentiation. 148 ORDINARY DIFFERENTIAL EQUATIONS. To this end let a and b be any two constants. Then v — au — b = (4) is a differential equation of the lirst order which is invariant under U"f, since U"{v-au-b) = 0. If now a is supposed to retain a fixed value, while b varies, (4) represents oo^ differential equations of the first order which are invariant under Iff. Each of these differential equations represents oo^ integral curves in the plane, so that there are oo^ families, each of oo^ curves, which as a system are invariant. This system of oo ^ curves must be represented by a diflFerential equation of the second order, which must also be invariant under U"f. We obtain this differential equation of the second order by differentiating ^(35. y,y')-au(x,y) = b totally with respect to x. Hence, we find, dv du - ... dS-'^dS=0' (5) ^yy'y -a = 0; (5') or, briefly, w(x, y,y',y")-a = (6) Since (6) is invariant under the G^, we must have U"(iv-a) = 0, by means of w — a = 0. But, since a is a constant, U"iiv-a)^U"{w), that is, U"iw) = 0; and w is a solution of the linear partial differential equation U"f= 0. LIE'S EQUATIONS OF SECOND ORDER. 149 From (5') we see that since v must contain y', w must also contain y" ; hence we can take this function to be the third solution of ^"/=0, that is, the third integral of the system (3). It is seen further from (5) that dv so that the most general invariant differential equation of the second order may always be written in the form 108. The complete integration of (7) may be accom- plished by the integration of an ordinary differential equation of the first order in two variables, together with one quadrature. For (7), as is seen by its form, may be considered a differential equation of the first order in the variables u and v, and if (7) has been integrated, say in the form v = dy it is seen at once that we may write dv u = x, v^y, w = ^ = y . Thus all differential equations of the second order, F(x,y',y") = which do not contain the variable y, are invariant under the translations along the y-axia. If F = be written in the form y"-^(x,y') = 0, dv ^, , . or ^-*<^' '^) = 0. M + C», CLASSES OF LIE'S EQUATIONS OF SECOND ORDER. 153 it is seen that this is a differential equation of the first order in u and v. Its integration -will give an equation of the form v = \lr(u,Ci), or y' = yfr{x, c^), so that a quadrature will give the general integral required in the form y=Ylr{x,c^da. Excmiple. Given (l-x^)y-^-2 = 0. This equation, not containing the variable y, admits of "'■% Hence we must substitute SO that there results (l—vP\—. — w» - 2 = 0, ^ 'du, ' dv u 2 or izl — J, z =0 du 1 — M^ ■ 1—u^ This differential equation of the first order in u and v is linear ; its general integral is given by Art. 68 in the form Vl-M^ 2 or y=-^==(8in-»x+ci). This equation in x and y must also admit of Uf, Art. 108, so that another quadrature gives the required general integral in the form y = (sin~':c)2 + 2cisin-'a: + Cj. 112. It may be remarked that the equation y"=^{y'). (1) that is, the general differential equation of the second order in which neither of the variables is present, admits of both of the O^'s 154 ORDINARY DIFFERENTIAL EQUATIONS. We find from (1) by a quadrature 1^) ""'"'■'" (Ci = const.) or, say y' = w{x+c^); whence, by a second quadrature, y= ywix+c-^dx+c^ 113. To find cM differ entiaZ equations of the second order which, are invariant under the G^ of affine trains- formations The twice-extended Q^ is while the simultaneous system to be integrated is dx_dy _ dy' _ dy" ~^-lS'-^'--2y"- It is evident that we may assume, Art. 63, u = y, v=xy'; so that ^^^^ xdy'+y'dx ^xT+l^ du ay y The required differential equation of the second order has therefore the form xy"+y' y ■^{y,!^y')=^. ^-^iy,xy') = 0. By integrating this differential equation of the first order, a resvdt of the form xy'—w{y,Ci)=0 (Ci = con8t.) CLASSES OF LIE'S EQUATIONS OF SECOND ORDER 166 is found ; and since this last diflFerential equation of the first order is known, Art. 108, to be invariant under a quadrature will give the general integral required in the form y = Q{x, Cj, C2) = 0. (Cg = const.) 114. In an analogous manner the most general differ- ential equation of the second order which is invariant under the Q^ since u = x,v=-, will be found to have the form y It should be noticed that the so-called abridged linear equation of the second order of the form y"+X^{x)y'+X{x)y = 0, where X^ and X are functions of x alone, is a particular case of the general differential equation of the second order which is invariant under For the above invariant equation may obviously be written y ^ y^ 156 ORDINARY DIFFERENTIAL EQUATIONS. when we assume If now we suppose "^ to have the special form the invariant equation will assume the form of the abridged linear equation. A further discussion of this equation will be given in Chapter XL 115. To find all differential eqwations of the second order which are invariant under the O-y Here it is readily seen that , , dv „ „ u = x, v=,j>y -y, ^ = ^y - y, where ' and <{>" are written for dx' da? respectively. Thus the most general invariant differential equation is y" - "y - *(a'. y' - 'y) = o- If ^ is an integral-function of the abridged linear equation of the second order y"+X,{x)y'+X,(x)y = 0, that is, if (p satisfies the identity ^"+X,(x)^'+Zi(a!)0 = O; then the general linear equation of the second order y"+X^ix)y'+X,{x)y+X,(x) = CLASSES OF LIE'S EQUATIONS OF SECOND ORDER. 157 will be a particular case of the above invariant differential equation. For we only need assume $ in the form when the invariant equation becomes y" + Xlx)y'-{£±^)y + X, = 0- that is, since satisfies the abridged equation y"^Xp^) y'+X^{x)y + X,{x) = 0. 116. To find all differential equations of the second order which are invariant under the G^ of perspective transformations TTf y , 3/ •' ?)X ^2iy It is seen from Art. 64 that we may here assume V ^x' '^^y' ,, , dv xy" so that T- = — - — du , y y-l Thus the most general invariant differential equation of the second order is ^-*e.o=»^ " X or, as it may obviously be written, 158 ORDINARY DIFFERENTIAL EQUATIONS. The integration of the differential equation of the first order in it = - and v = y', f-*e^)-». X will give a result of the form 2/'-^(|.c) = 0. This equation must, of course, admit of Uf, and hence its general integral, and thus the general integral of F = 0, may now be found by a quadrature. Example. Given the differential equation This is obviously an equation of the form F=0. Hence it may be written in the form _. ,, / \ and, in fact, we have -^L ^=0. y_y y dv _v _ du~v, ' that is, v=c,M, X A quadrature will now give the general integral required in the 117. The values of u and v for the groups 3/, 3/ 3/ 3/ 3/ 3/ „3/, 3/ are given Arts. 66-73. It will be a valuable exercise for the reader to find the corresponding invariant differential equations, or Lie's equations, of the second order. Hence CLASSES OF LIE'S EQUATIONS OF SECOND ORDER. 159 118. The criterion that a given differential equation of the second order, Q(x,y,y',y'') = 0, shall admit of a given twice-extended Oj^, is that the expression shall be zero, either identically or by means of f2 = 0. It is often possible to find, from this condition, the Gj of which the given equation admits, as is best illustrated by an example. Example. The condition that the equation shall admit of Uf, ia that the expression u"(a) = - ^yy + »/y" - ij'^ry - sy v + W - ^v" shall be zero identically, or by means of 12 = 0. On comparing this expression with 12=0, we see that for ^ = 0, »; =y, and thus rf=y', Y's/,wehave Z7"(fl)s2ft. The given equation must therefore admit of y^, and hence be one of the type M^-*(x, ^)=0; and, in fact, it may be written Now, Art. 114, assume u=x, »=^i and the last equation takes the form dv v^(Z + uv) du 1-MW ' or {l-uv)udv-(3uv+uh^)vdu=0, 160 ORDINARY DIFFERENTIAL EQUATIONS. an equation of the first order which may be integrated, Art. 67, by a quadrature. Since the resulting equation will admit of the Gi, Uf^y 3/ a second quadrature will give us the general integral of the given differential equation of the second order, 12 = 0. 119. We shall apply the theory of integration of this paragraph to an example involving geometrical con- siderations. A family of curves in the plane is often defined by an equation expressing a relation between such magnitudes as the subtangents, the radii vectores, the perpendiculars from the origin on the tangents, etc., giving rise by that means to a differential equation of a certain order. For example, suppose it is required to find all curves which are defined by a relation between the line r connecting the point {x, y) with the origin ; the angle \^ between this line and the radius of curvature, p; and the radius of curvature itself. Such a relation is given by an equation of the form *(r-, i/r, p) = 0. It is geometrically evident that such a curve will pass, by means of a rotation, into a congruent one. In other CLASSES OF LIE'S EQUATIONS OF SECOND ORDER. 161 words, the family of curves represented by # = admit of the G^ of rotations where, of course, the values of r, i//-, and p, in $ = 0, are to be expressed in Cartesian coordinates. Since, in the variables x and y, $ = will be a differ- ential equation of the second order, and since we know the path-curves of Uf, it is clear that, by Art. 108, the above family of oo^ curves may be determined by the integration of a differential equation of the first order, together with a quadrature. 120. In integrating the following differential equations of the second order, the equations should fii-st be examined to see whether they are eocact or not, Art. 103. If an equation is exact, its integration may, Art. 103, be reduced to the integration of a differential equation of the first order. If the differential equation of the second order is not exact, it should be compared with the types of invariant equations. Arts. 110-117 ; and if the equation belongs to one of those types, its integration is to be accomplished by the method indicated for that type. If the equation is not exact, and does not belong to one of the given types, the G^ of which it admits is to be sought by the method of Art. 118. In examples (l)-(25) it should be verified geometrically, whenever practicable, that the family of cc^ integral curves found admit of the G^ which was used in integrating the differential equation. Thus, it is geo- metrically evident that the integral curves x^+2c^x+y^+2c2y = of Ex. 26, which are the oo^ circles through the origin, admit of the Gj of rotations. We give, for convenience of reference, a table of some of the more important invariant differential equations of the second order. 162 ORDINARY DIFFERENTIAL EQUATIONS. Oroup of One Parameter. Type of Invariant Differential Equation. (^^^^^2' (i)F(y,y,y')=o. This invariant equation includes, of course, the types This equation includes F(^,/)=0, F(y',/)=0. (3) jy^^g, (3)F(y,^y,^')=0. (4)r/.,| (4)f(.,^,^^) = 0. (^)^^=^I+4 (5)F(|,y,^")=o. EXAMPLES. (1) yy'+y^ = l- (2) (l-x')i/"-j;^'+2/=Q. (3) y"=xe'. (4) y=^ + sin.2?. (7) ff^a. (8) y' = -^. (9)ay'2=i+y2. (io)y=y2+i. (11) y'= -(yni). (12) aY''={\^^f. (13) :j^" +y = 0. (14) y - xy'' =f{y"). (15) (H-a:2)y' + l+y' = 0- (16) {\ - x')/ + xy' = x. (17) m/'+^=3f. (18) (a2-:,:2)y'-^y + ^ = 0. (19) y'+yy=0. (20) y(l-logy)y'+(l+logy)y2=0. (21) y = ^. (22) xy" - xy"^ +y = 0. (23) ihf - xy' - 3y = 0. (24) m/'^ + xyf - yy' = 0. (25) xY-xy'+y = 0. (26) (^2 +y)y' - "ixy'^ + 2yy ^ - 2:t:y' + 2y = 0. CLASSES OF LIE'S EQUATIONS OF SECOND ORDER. 163 The following geometrical examples lead to ordinary diflferential equations of the second order which admit of a Gy The formulae for a plane curve given, Art. 73, together with the two given below, are convenient for reference. Diflferential of an arc s, measured fro:n a fixed point on the curve up to a point (x, y), Badius of Curvature, x „ ■ y (27) Determine a curve such that the length of the arc measured from a fixed point on it is equal to the intercept of the taugent on the axis of x. (28) Show that the curve whose radius of curvature is proportional to the cube of the normal is a conic section. (29) Required the family of curves in which the radius of curvature is constant and equal to a. (30) Determine the family of curves in which the radius of curva- ture is equal to the normal (a) when the two have the same direction, (6) when the two have opposite directions. (31) Determine the family of curves in which the radius of curva- ture is equal to twice the normal (a) when the two have the same direction, (6) when the two have opposite directions. (32) Show that if t is the angle which the- tangent at any point {x, y) on a given curve makes with the a:-axis (see figure, Art. 119), while p is the radius of curvature at that point, all curves defined by a relation of the form Q,{x, T, p)=0 may be determined by the integration of a diflTerential equation of the first order and a quadrature. (33) If the angle between the x-axis and the line joining the centre of curvature with the origin be designated by 6, and if <^ have the meaning of Art. 119, show that all curves defined by a relation of the form fi(<^, T, e) = o between the three angles 4>, r, d may be determined by the integration of a differential equation of the first order and a quadrature. CHAPTER X. THE DIFFERENTIAL EQUATION OF THE m" ORDER IN TWO VARIABLES. 121. In this chapter we propose to indicate briefly how the methods of Chapters IV. and IX. for integrating invariant differential equations of the first and second orders may be extended to equations of an order higher than the second. SECTION I. Lie's Differential Equations of the m'* Order. 122. A differential equation of the m** order in two variables has the general form n{x,y,y' 2/<"'>) = 0, where fi must, of course, actually contain y^'"^ ; and the complete primitive, or general integral, (Art. 5), of this equation has the general form •^{x,y,ci, ...,c„) = 0, where Ci, . . . , c^ are m independent arbitrary constants. We saw iu Art. 72 that the general integral of an ordinary differential equation of the first order in two variables may always be expressed in the form of an infinite series involving one arbitrary constant. We shall now show that the general integral of the ordinary differential equation of the m" order in two variables may, similarly, be expressed as an infinite series involving m HE'S EQUATIONS OF THE to" an&f!R. 165 arbitrary constants — although, as in Art. 72, we shall not investi- gate the question of whether this series always converges or not. It may be remarked that this method for obtaining the general integral is of \itt\G practical value for such equations as are neither linear (Chap. XL) nor reducible to a linear form. If y'»'=F,(x,y,/,...y"-") (1) be the given differential equation, by differentiation we find y™*" as a function of x and y, and the differential coefficients up to the m", if Fj actually contains y™"". Substitutii>g, in the result, fory*" its value as given by (I), we find for y"*", y-^^^F^c^, y, y, ...y™-") (2) Differentiating (2), and reducing as before, we find y»+«=F3(^, y, y, ...y— ") ; (3) and proceeding in this way we see that all differential coefficients of y of an order higher than the to" are expressible in terms of X, y, y, ..., y™"", by means of (1). Now when we assign to x the initial value .i^, let the correspond- ing initial values of y, V, . . ., y*"-" be represented by y^ y^, . . .,yo"°~", while the second members of (1), (2), (3), ... become F/"', Fj™', Fa'"*, ... req)ectively ; then, by Taylor's theorem, we find, as in Art. 72, (m—ii V (0) ■p (0) In this expression for the general integral of (1), x^ is a special numerical value of x ; and, as the F/"' are functions of the to arbitrary constants y^, y'„, ... y^"""'', (4) really contains only m inde- pendent arbitrary constants. 123. We shall now give a method for finding all differential equations of the m*'' order, ft = 0, which are invariant under a given G^. Such equations are some- times designated as " Lie's eqtuatians of the m'* order." In order to find all differential equations of the second order which are invariant under a given G^ 166 ORDINARY DIFFERENTIAL EQUATIONS. we saw that, if represent the twice-extended G-^, it is necessary to find three independent solutions of the linear partial equation in four variables £/"'/= 0. If these solutions are represented by w(a;, 2/), v{x,y,y'), ^ = w(x,y,y',y"), then the roost general invariant differential equation of the second order is F{u,v,w)=0. In an entirely analogous manner we may see that to find the most general differential equation of the third order which is invariant under Uf, it is necessary to find four independent solutions of the linear partial differ- ential equation in five variables where U"'f is the thrice-extended G^, so that Just as we saw in Art. 107 that we could assume dv so now, if w, V, vj be three solutions of t/"'/=0, the function dw _ dH du ~ du^ may be seen to be a fourth solution of U"'f= 0, which must contain y'". Hence the most general invariant LIE'S EQUATIONS OF THE m'" ORDER. 167 differential equation of the third order has the form Unless the path-curves of the G^ u = const. are known, it will be necessary to integrate a differential equation of the first order in two variables dx_dy to find u. Then v, Art. 56, may be found by a quadrature, and of course the other two solutions dv d^v du' du^ by mere differentiations. It is obvious now that the most general invariant differential equation of the m}^ order will have the form „/ dv d^v d™-iv\ „ and it is clear that, in the most unfavourable case, this differential equation may be established by the integra- tion of a differential equation of the first order in two variables, a quadrature, and (m — 1) differentiations. Since u and v are given. Arts. 62-68, for the G^ of those Articles, of course the corresponding invariant differential equations of the m''' order may be found, as indicated above, by mere differentiations. In accordance with the definitions of differential in- variants of the first and second orders, given Arts, 57 and 106, we now define the function dv} as a differential invariant of the (i-|-l)'* order of the G,Uf. 168 ORDINARY DIFFERENTIAL EQUATIONS. 124. By Arts. 42 and 43 it is clear that the necessary and sufficient condition that a given differential equation of the m"' order, Q(x,y,y', ...3/<-)) = 0, shall be invariant under a given Gj Uf, is that the expression shall be zero, either identically, or by means of J2 = 0, — where is the m-times extended G^. In order to reduce the problem of integrating 12 = 0, we must first find u, by integrating, if necessary, dx _dy Then v may be found by a quadrature ; and the above differential equation of the m"" order may be put into the form „/ dv d'^-'^v\ - or, if we choose. du^-^ .$ / dv d"'-^v\ „ This is a differential equation of the (m— l)**" order in u and V. If its general integral has been found in the form ^ -/(«-, ci, Cj, ... Cm_i) = 0, the last equation will be a differential equation of the first order in x and y, which, by Art. 42, must admit of the given G^ Uf; and hence the general integral of fl = may be finally found by a quadrature. CLASSES OF LIE'S EQUATIONS OF m" ORDER. 169 SECTION II. Classes of Lie's Differential Equations of the m"' Order. 125. We shall now illustrate the method given in the last section for finding classes of invariant differential equations of the m"" order by some of the simplest possible examples. 126. To find all differential equations of the m'* o^'der which are invariant under the 0^ of translations along the x-axis, ._3/ ^/-i- By Art. 62, we have dv y" u^y,v^y,w^^^y. dw^ d^v __ y"'y'-y"^ ^^ dv, ~ dv? ~ y'^ ' ' Thus the most general invariant equation of the m*^ order may obviously be written vf,j V y" y"'y'-y"" Vo- or, in the equivalent form, Hence, an equation of the m** order, 12 = 0, which is free of x, admits of the G^, Uf; and may always be written as an equation of the (m — l)**" order in the variables u = y,v = y', in the form dv d'^-'^v^ J dv d'^-^v\ f. V'''''d^'- d¥^) = ^ 170 ORDINARY DIFFERENTIAL EQUATIONS. 127. To find all differential equations of the m"' order which are invariant under the G^ of all translations along the y-axis, Here, by Art. 62, we have dv dw d^v ,„ , d}j,^d^^^y ' ^^- Hence, in this case, the most general invariant diflferential equation of the m"^ order has the form F(a;, 2/', /',... i/('»)) = 0; and it is obvious that it may be written as an equation of the (m — 1)*^ order in the form dv d ( dv d'^-H\ „ 128. To find all differential equations of the m}^ order which are invariant under the G^ Here we have, Art. 63, u=., v=y-; wJj^^yy^.y--(y)\ y. du 2/2 y \yy ^.^=2L:_3^^' + 2pY, etc. du du^ y y y \y/ Hence the most general invariant differential equation of the m* order may be written ^ y y \y^ y y y ^y^ ' CLASSES OF LIE'S EQUATIONS OF m'" ORDEE. 171 or in the equivalent form y' y" y'" j/'"'^ f2 a;,^,^, ^,...^- =0. ^ y y y y ^ Thus every equation of the form fi = 0, of the m"' order in x and y, may be written in the form of an equation of the (m — I)"' order in u and v, by making the substitutions indicated above. It is seen that the so-called abridged linear equation of the 7)1*'' order, of the form is a particular case of the equation fl = 0. 129. To find all differential equations of the m'* order tvhich are invariant under the G^ Here, by Art. 115, we have for u and v the forms dv du' u = x, v=^y'-y"-"y, ^2^y"'-ry+'y"-"y'- etc. Hence, the most general invariant differential equation, of the m**" order may be written F(x, .py' - 4,'y, y"' - "'y + is an integral-function of the corresponding abridged linear equation. The proof is precisely analogous to that of Art. 115 for equations of the second order. 130. It will be very valuable exercise for the reader to find the differential equations of the 3"*, i"", . . . orders, which are invariant under the simple O^s given in Art. 117. 131. It may be noticed that the simplest form of differential equation of the m"" order that is invariant under the 0^ is 2/('»)=Z(a;), (1) where Z is a function of x alone. It is obvious that the general integral of this differential equation of the m*'' order may be found by m successive quadratures. 132. It is clear that when the equation of the type of Art. 126 has the special form its integration may be facilitated by assuming ^'•^ = z. The equation fi = is then of only the (m — %f^ order in the variables x and z, dz d^-'z V'dx' - da;"'-/""' and is of the type of Art. 126 still. Its integration will give a result of the form z = 'jP = ^{x,c, ... c„-.); so that, by the preceding article, the general integral required may now be found by i successive quadratures. GLASSES OF LIE'S EQUATIONS OF m'" ORDER. 173 This method is particularly applicable when i2 = has one of the simple forms fi^2,(m)_/(y(m-l)) = 0, (2) or n = i/('»)-/(t/('»-2)) = (3) The first equation, when we put 3/<'"-i) = 2;, becomes dz . and may be integrated by a quadrature giving z as a fvmction of x and one constant. The second equation, when the same substitution is made, becomes dP'z for the integration of which a method is given in Art. 110, by means of which z is found as a function of x and two constants. Example 1. Given y_?/" = v/l+«"2. This is an example of equation (2), Art 132. Hence, assume y = z, and -we find whence x=c-\-Jl+z\ (c = const. ) Thus, solving for z, we have z = 'J{x — cf — \, or y'' = >J{x — cf—\. By two successive quadi-atures the general integral may now be found. Example 2. Given ay'=y. This is an example of equation (3), Art. 132. If we write s for y", the equation becomes v?;!' =z. By Art. 110, we find from this * _« 2 = 0,6" +C2« ". 174 ORDINARY DIFFERENTIAL EQUATIONS. X X Hence, from y" = Cje" + c^ », by two successive quadratures, we find the general integral X I EXAMPLES. Integrate the following differential equations : (1) xy"' = 2. (2) af'=y\ (3) f'=-. {4) y"=xcosx. (5) a;y = 2y'. (6) ?'/■■ + 4 cos .r = 0. (7)y-'=^e'. (8) /" = sin'a:. (9) y"=l+cos.r. (10) y"=y(i+y). (ii) ixy"y=y"'^-a\ (12) f'y'^=\. (13) xf'+zy"'=o. (14) y'y'=(i-y")(i+y'2)l CHAPTER XL THE GENERAL LINEAR DIFFERENTIAL EQUATION IN TWO VARIABLES. 133. In the first section of the present chapter we shall give a method for finding, by mere algebraic operations, the general integral of the ordinary linear differential equation of the vi^^ order with constant coefficients and the second member zero. In the second section we shall give methods for the same equation, when the second member is not zero. In the third section we shall show how the knowledge of the fact that the general linear differential equation is always invariant under a known Gj^ may be used to lower the order of the equation. SECTION I. The Abndged Linear Equation of the m'* Order, with Constant Coefficients. 134. The differential equation of the m*'' order of the form 3/'»)+X„-i(a;)2/(-^)+...+X,(a^)2/ = X(cc) (1) is known as the general linear differential equation of the m"" order. We reserve for the third section a treatment of this equation when the Xi are functions of 176 ORDINARY DIFFERENTIAL EQUATIONS. X ; for the present we assume that the Xi are all constants, so that (1) may be written 2/'»)+^„.i2/<™-i)+... + Ji3/ = Z(a;) (2) The last equation is known as the general linear equation of the m*'' order with constant coejficients. If, in particular, X(x) = 0, the equation (2) becomes y(^)+A„,.,y<^^-^>+...+A^y'+A^y = 0, (3) which is called the abridged linear equation corresponding to (2). We shall discuss equation (3) in this section. 135. Although, as we know from Art. 128, (3) is invariant under the 0^ Uf^y% 80 that the integration of (3) may be reduced by the method of that article, a more expeditious way of finding the general integral, and one not involving any processes of integration, will now be explained. To this end, substitute in (3) for y the value where a is a constant to be determined. It is seen that each of the terms of (3) will be multiplied by the factor e"*, which may therefore be discarded, so that we have a-^+Ara.ia^-'^ + ...+A^a + A^ = Q (4) This is an algebraic equation of the m**" degree in terms of a; and for each root, ai, of this equation, it is clear that we have a corresponding particular integral of (3) of the form 2/i = e"^*. Thus, if ftj, aj, ... «„ be the m roots of (4), the equation 2/ = Cie»i»+C2e«^+...+c„e«-.*, (5) ABRIDGED LINEAR EQUATION OF m^ ORDER. VJ*I will be the general integral of (3), Art. 122. For this equation contains m independent arbitrary constants, and the value of y given by (5), when substituted in (3), satisfies (3) identically. Example. Given the abridged linear equation of the second order, y-5y' + 6y = 0. The corresponding algebraic equation of the second degree is ■with the roots 2 and 3. The general integral of the differential equation is therefore, by (5), That this is correct may be immediately verified. 136. In the case when (4) has a double root, say the equation (5) no longer represents the general integral of (3). For in that case the first two terms in (5) reduce to the form (Ci + Ca).^'^ where Ci + Cj may obviously be replaced by a single arbitrary constant c; and since (5) now only contains (?n — 1) independent arbitrary constants, it is no longer the general integral of (3). In order to obtain the general integral, let us suppose that the above-mentioned two roots are not exactly equal, but that they difier by a quantity k, which will ultimately be made to vanish. The part of (5) depending upon the roots aj and a^ will then have the form c^e'"»^+C2e('"+''>* (6) {kx^ 1 (Ci + Cj) + Cj/ca; -)- CgK -Tg-I- . . . |- 178 ORDINARY DIFFERENTIAL EQUATIONS. Since Cj and c^ are arbitrary, we may assume them to be infinite in such manner that, as k approaches zero, G^K approaches a finite quantity B^, while Cj and c^ are taken with opposite signs, in such manner that c-^ + c^ is finite and equal to B^. Thus the sum (6) has the form e-^-[B,+B^+B,'^+...y so that, when k = 0, (6) becomes e'^{B, + B^}. Thus we see that in the case when (4) has a double root ai = a2, the arbitrary constant (G-^ + C^) must be replaced in (5) by a binomial expression of the form (B,+B^). In an entirely analogous manner it may be shown that if (4) has an r-fold root, the r terms coalescing in (5) must be replaced by a polynomial of the form e''i^{B^ + B2X+...+BrX'--'}. Example. The algebraic equation corresponding to y-6y+9^=o has the double root 3. Hence the general integral of the differ- ential equation is ^ ^ ^(^^ + ^^y 137. When (4) has a pair of imaginary roots, the corresponding constants of integration are to be assumed imaginary in order that the pair of terms in (5) may be reduced to a real form. Thus, if be a pair of imaginary roots, the corresponding terms in (5) are = e'"[(ci + Cj) cos/3x + s/ — I. (Ci — c^ sin ^x\. LINEAR EQUATION OF m" ORDER. 179 If now Cj and Cj be considered imaginary, and if we assume the real form sought will be e<^{Acosfix+Bsmpx) (7) It is readily seen, as in Art. 136, that if a pair of r-fold imaginary roots occurs in (4), each of the arbitrary con- stants in (7) must be replaced by a polynomial of the (r — 1)*'' degree of the form Example. Given y - ey + 1 3y = 0. The corresponding algebraic equation, a2-6a + 13 = 0, has the pair of imaginary roots, a, = 3 + 2\^-l, a2=3-2V-l. Hence, by (7), the general integral is seen to be y = ^{A cos 2x+B sin 2x). SECTION II. The Linear Equation of the m'* Order, with Constant Coeffbdenta and the Second Member a Function of x. 138. The problem of finding the general integral of the equation 3/W+^„.i2/'"-^)-|-...+A3/ = ^(a5) (1) is intimately connected with that of finding the general integral of the corresponding abridged equation 2/W+^m-i3/('"-'^+---+^3/ = (2) For suppose that the general integral of (2) has been found in the form y = c^d^<'+c^e'^-\-... + c^d^ (3) 180 ORDINARY DIFFERENTIAL EQUATIONS. and that ^* B^+B^e^ is a particular integral-function of (9). By substituting y = B^-^B^^ in (9), we find 536" -H454e2z + 853«'-f- 16546=^ -t-ie^ae^-l- 16546^ = 4e»-e2' ; that is, B,^± B,^-^. LINEAR EQUATION OF m" ORDER. 183 Thus we find for (9) the general integral 140. A second method for finding a particular integral- function of (1) is that which is commonly known as the " Variation of Parameters." To find <}>{x) (Art. 138), by this method, we first find the general integral of (2) in the form (3) ; then, considering the m arbitrary constants as variable parameters, by substituting the value of y given by (3) in (1), we determine the parameters in such manner that (1) is satisfied. The m parameters may evidently be subjected to (m — 1) arbitrary conditions; and the system of con- ditions which produces the simplest result is that which demands that all the derivatives of y of an order lower than the m"" shall have the same values when the parameters are considered as variables that they have when the parameters are considered as constants. Example. Given y" + n'y = X{x) (11) The general integral of the abridged equation corresponding to (11) "^ y=CiCoanx + C2SVD.nx. (12) Now, supposing Cj and Cj to be variables, we wish to determine these quantities in the simplest manner possible, so that (12) will be the general integral of (11). Differentiating (12), we have , . dcs , . dc« y= -nCi%\rinx+ncj^cosnx+cosnx--r^+smnx--T^ ; and thus, in order that y' may have the same value as if c^ and Cj were constants, we must have cosnx^ + smnx^=0. (13) dx dx Also, differentiating the equation y'= —nci sin nx + nc^ cos nx again, we find dc, dco y = - 7i^(cj cos nx+c^ sm nx) -namnx-j^+ncoanx-^. In order that y shall satisfy (11) we must have, therefore, -nBinnx-T^+ncoanx-j^=X{x) ; (14) 184 ORDINARY DIFFERENTIAL EQUATIONS. and from (13) and (14) we find dc, 1^ . dcif ^ — n-T^ = X aiTi. Tix, n^ = X coSTix. ax ax Hence, by two quadratures, Ci= — [XAn.nxdx-'ra^, <^i = - j Xcosnxdx+a^ ; 71 J 11 J and the general integral of (11) is y = — cosnx j Xsinnxdx+ - sin nx I Xcoanxdx + aicoatuc + a^sinnx. It will be seen that the same result may be obtained directly by Art. 146. 141. It should be noticed that all equations of the form (a+6a;)'"2/W+^i(a+6x)'"-Y'»-i)+... + A„,.-^(a+bx)y'+A^y+X{x) = (15) may be transformed into linear equations with constant coefficients by the simple substitution a + bx = e', t being the new independent variable. If, in equation (15), the constant a happens to be zero, (15) is called the general homogeneous linear eqvMtion of the m"' order. Example. The equation (a + 6.r)y' + ^i(a + 6a;)y'+J2y = 0, when we assume a + bx=^, becomes linear with constant coefficients. For we have y-dx~^^ dt' so that the above equation becomes 6^g-(6^-^.6)| + 4^ = 0, an equation of the form (2). GENERAL LINEAR EQUATION OF m'" ORDER. 185 SECTION III. The General Linear Equation of the m'* Order in which the Coefficients are Functions of x. 142. It was shown, Art. 129, that the linear equation of the m*'' order 2/W+X„_i(x)y(— i)+...+Z,(;r)2/ + Z„(x) = 0, ...(1) where the Xi are functions of x alone, is invariant under the Gj if is any function of x which satisfies the abridged linear equation corresponding to (1) 2/W + Z^.i(a3)i/('»-i)+... + X/a;)2/ = (2) We shall call (pipe), under these circumstances, a pariicuZar integral-function of (2). 143. In order to depress the order of equation (1), we know, Art. 129, that we must make the substitutions , V ^+{"+X^i>'+X^4,}y = 0; or, since satisfies (2'), ^+X,v+X, = 0, X^, Xg, and being expressed as functions of u. The general integral of this linear equation of the first order is. Art. 68, or, restoring the variables x and y, ^y'-^y = e-\^'''''^\-\^Xlx)^x)e\''*'''\dx^c}^. GENERAL LINEAR EQUATION OF m" ORDER. 187 The general integral of this linear equation of the first order, that is, the general integral of (1') is. Art. 68, 145. The abridged linear differential equation of the m*^ order (2) admits of the G^ so that the order of (2) may always be depressed by unity by the method of Art. 128 ; but the resulting differential equation of the {tn — Vf^ order in the vari- ables u and V is usually not linear. 146. If, in particular, we assume that (2) has constant coefficients, and is only of the second order, of the form y"^-Ay'->rBy = (i, (^, 5 const.) written in u and v, by Art. 128, it becomes an equation of the first order which may obviously be integrated by a quadrature, when another quadrature will give the general integral of the differential equation of the second order. In this manner the particular integral-function ^(x) may be found, — that is, by ascrib- ing any numerical values desired to the two arbitrary constants in the value of y found. Also, ^(x) may be found by Art. 135. When ^(x) is known, the general linear equation of the second order, with constant coefficients may, by Art. 129, be written as a linear differential equation of the first order in u and v. The last equation may be integrated by a quadrature, Art. 68; when a 188 ORDINARY DIFFERENTIAL EQUATIONS. second quadrature will give the general integral of the general linear equation of the second order with constant coefficients. Hence, since ^(a;) may always be found, by Art. 135, by algebraic operations, we see that the linear differential equation of the second order with constant coefficients may always he integrated by two quadratures. For practical work, however, the method of Art. 139 will usually be found more advantageous. Example. Given the diflferential equation y" + nh/ = co&7ix (3) It is seen that sin nx is a particular integral-function of the abridged equation f + nhj=0. Hence the above differential equation of the second order admits of -Q, Uf= sin nx^ ; and to depress the order of the equation we have. Art. 129, to substitute in (3), . V , J, 1 dv . aiunx aiunx du ^ We find -; =- = cos nu. Bin nu an or dv = sin nu cos nudu. XT sin^nM / . , Hence v = —z: h c, , (Cj = const.) or, since v = sin mn/ - n cos nxy, u = x, we havBj sin nxy' - n cos nxy= — 1- Cj. The integral of this linear differential equation of the first order may, by Art. 68, be found by a quadrature in the form X sin n^ c, cos nx "in n ' GENERAL LINEAR EQUATION OF m" ORDER. 189 EXAMPLES. Integrate the following abridged linear equations with constant coefficients : (1) y"-7y + 12y = 0. (2) 3y-l(Y + 33/ = 0. (3) y"-4y=o. (4) y"-7y+6y=o. (5) y - 12y + 27y = 0. (6) y" - 4/" + Gy" - 4/ +«/ = 0. (7) y'-4a6y + (a2 + 62)V=0. (8) y"+y"+y-3y = 0. (9) y + 2y' - 8^^ = 0. (10) y"' - sy + 4y = 0. (11) y'+2mY'+?tV = 0. (12) y''-3y" + 3y'-y = 0. Integrate the following linear equations with constant coefficients and the second members functions of x : (13) y - 7y + 12y =^. (14) f - 2y"' + If - 2/ +2, = 1. (15) y"-2y+y=e'. (le) y'+?i2y=i+:c+r2. (17) f-y+y=e'. (18) y'-3y+2y=xe'« (19) y ' + 4y = a; sin^.^:. (20) y'" - 2y + 4y = e^cos x. Integrate the following equations by the method of Art. 141 : (21) 3fly"-xi/-Zy=^0. (22) {x+afy'-A{x + a)i/ + &y=0. (23) a?y" -xy' + iy=x\ogx. (24) (2j;-l)y"+(2^-l)y-2y=0. Integrate the following equations by the method of Sec. III. : (25) {\-x^)j/'+xj/-y=x{\-x^y. (26) i/' -xr/+{x-\)y=a^. (27) aY'+4ay'+2y=e'. (28) xy''+y=x. (29) (l-x2)y'-^-2/=0. (30) /-^y+ jrT2'=^-l- Other examples for practice may be found in the Examples at the ends of Chapters IX. and X. CHAPTER XII. METHODS FOR THE INTEGRATION OF THE SIMULTANEOUS SYSTEM. 147. In the first section of this chapter we shall give briefly the simplest of the methods which do not involve transformation groups for integrating certain forms of simultaneous systems of ordinary differential equations. In the second section we shall give a general method of integration for a simultaneous system in three vari- ables, when the equivalent linear partial differential equation of the first order in three variables admits of a known 0-^; while in the third section we shall give an application of the theory developed in the second section to ordinary differential equations of the second order in two variables. SECTION I. Special Methods for Integrating Certain Forms of Simultaneous Systems. 148. In Art. 23, Chap. II., we gave a method for inte- grating a simultaneous system of the form dx _ dy _ dz X(^)-Y(^)- Z{x, y, z) ' that is, we saw that when the first equation had been integrated, — by the methods of Chapter IV., — either x or INTEGRATION OF SIMULTANEOUS SYSTEMS. 191 y might be eliminated from Z, so that a second integral of the simultaneous system might be found by integrating a second differential equation of the first order in two variables. It is obvious, therefore, that a simultaneous system of the above form may be completely integrated by integrating two ordinary differential equations of the first order in two variables. 149. The general form of the simultaneous system in three variables is dx_dy _dz ,^, where X, Y, and Z are usually functions of all three variables x, y, z. We may write the ratios (1) dx _dy _dz _Xdx + ixdy + vdz , , 'X~T~~Z~ \X+fxY+vZ ' ^ > where X, fx, v may be either constants or functions of the variables. If it is possible to choose \, fi, v in such manner that 'KX+f,Y+vZ=0, then also \dx+ij.dy-{-vdz=Q; (3) and the integral-function of the total differential equa- tion (3), if it may be found by the methods of Chap. VIII., will obviously also be one of the integral-functions of (1). That is, if ^{x,y,z) = c is the integral of (3), it is also an integral of (1). The second integral of (1) may then be found by inte- grating an ordinary differential equation in two variables from, say, dx_dy X~T' when 2 has been ehminated from X and F by means of 192 ORDINARY DIFFERENTIAL EQUATIONS. Example. Given the simultaneous system dx _ dy dz mx -ny nx- Iz ly — mx' The method of the present article may be applied twice. If we choose A., ft-, V equal to I, m, n respectively, we find ldx+mdy + ndz = Q. If we choose X, /n, v equal to x,y, z respectively, we find xdx-\-ydy+zdz = Q. The integrals of these equations are obviously lx+my + nz=c^, and these two equations are the general integrals of the given system. For the geometrical meaning of the integrals of a simul- taneous system in three variables, see Art. 19. 150. The general simultaneous system in the (n + 1) variables Xi, ..., Xn, t, has, as we knov?, the form dx-i^ _ dx2 _ _ dxn _ dt . . 'X[-^^---^-T' ^^> where the X\, ..., X„, T, are usually functions of all the variables. If we choose any one of the variables, say t, as the independent variable, it will always be possible, by differentiating these equations a suflScient number of times, to eliminate all but one of the dependent variables and their differential coefficients. In fact, if no method for abbreviating the work suggests itself, we may always obtain, by differentiating each of the given equations (to — 1) times, exactly v? equations, which are just suffi- cient to eliminate {n — l) variables with their n(n — \) differential coefficients. The resulting differential equa- tion of the ■n."' order in two variables must then be integrated ; and from its general integral, and the system (4), the values of the other dependent variables may be found, giving a system of general integrals consisting, Art. 20, of n equations involving n arbitrary constants. INTEGRATION OF SIMULTANEOUS SYSTEMS. 193 Of course this method is most appropriate for the integration of systems of linear equations with constant coefficients, since we then have a definite method, Art. 135, for the integration of the system. 151. As an illustration of the preceding article, suppose that a system of two differential equations of the first order is given, connecting the variables x, y, and t, t being chosen as the independent variable. To find the equa- tion connecting x and t, we differentiate, if necessary, both of the given equations with respect to t; thus obtaining four equations connecting the quantities dx dy d^x d^y *' ^' di' di' W' W' from which we can eliminate y, -tt, -3^. The resulting equation will, of course, be a differential equation of the second order in x and t. The general integral of this equation will give x in terms of t and two arbitrary constants ; and by substitut- ing this value of x in one of the equations of the given system, y may be found. Example 1. Given the simultaneous system of linear equations, dx dy dt /ev Zx—y~x+y~\ These equations may be written and by differentiating the first we find, d^x dx dy_ dfi d^dt _ d^x ^dx , , . Hence __3^+^+y=0; or, from the first equation, d^x .dx , . . ___4^+4^=0. 194 ORDINARY DIFFERENTIAL EQUATIONS. The general integral of this linear differential equation of the second order, with constant coefficients, is found by Art. 136 to be x={Bi + Bit)e^ (6) Substituting this value of x in we find for y, y={B^-Bi->rB4)^ (7) Thus the equations (6) and (7) represent the system of general integrals of (5). Exam/pie 2. Given the system of linear equations, §-^+3^=6=', (8) in which the independent variable t occurs explicitly. Differentiat- ing the first equation, we have d^x dx dy _, d^^ ~di'^di~ By means of the equations (8) we may eliminate y and -^ from the last equation, giving By the method of Art. 139 the general integral of this equation is found to be x = {c^ + c^t)e-*' + ^^-^e^; and this value being substituted into the first of the equations (8) gives us at once, 152. It is clear that the differential equation of the second order, 2/"-ft)(x, y, 2/') = 0, may be regarded as equivalent to a simultaneous system INTEGRATION OF SIMULTANEOUS SYSTEMS. 195 of equations of the first order in three variables. For we have dx_dy _ dy' T~Y~oi,(x,y,y')' so that In an analogous manner it is clear that a diiferential equation of the m**" order in two variables is equivalent to a simultaneous system of m differential equations of the first order in (m + 1) variables. Similarly, a simultaneous system of difierential equa- tions of an order higher than the first may always be written as a simultaneous system of differential equations of the first order in the proper number of variables. For example, if in the simultaneoiis system of the second order d^_Y ^-v ^—7 dt^~ ' dt^~ ' dt^~ ' where X, Y, Z are certain functions of x, y, z, t, — we designate by x', y', ^ the differential coeflBcients, with respect to t, of x, y, and z respectively, — the above simul- taneous system may obviously be written, dao_ , dy_ , dz_ , dt~^' dt'^' dt~^' ^-Y ^-V —-7 dt~ ' dt~ ' dt~ Thus the simultaneous system of equations of the second order in four variables may be replaced by the simul- taneous system of equations of the first order in seven variables. If the six general integrals of this system, involving six arbitrary constants, have been found. Art. 150, the elimination of x', y', z between these integrals will give the three general integrals, involving six arbitrary con- 196 ORDINARY DIFFERENTIAL EQUATIONS. stante, of the simultaneous system of equations of the second order. 153. A method of integrating a simultaneous system of linear equations with constant coefficients and of an order higher than the first, analogous to that of Art. 150, will be sufficiently illustrated by the following example : Example. Given the system 5=7^+3y, (9) g=2.+6y. By differentiating the first equation twice we find and from this equation, by means of the equations (9), we find §-13g+36.=0. The general integral of this equation is, by Art. 136, x=(ci + C2t)e'' +(C3+ c^t)^ ; and by substituting this value of x in the first of equations (9), we find SECTION II. Theory of Integration of a Simultaneous System in Three Variables which is Invariant under a known 0^ 154. It was shown in Chapter II. that the general simultaneous system in three variables, of the form dx_dy _dz INTEGRATION OF SIMULTANEOUS SYSTEMS. 197 is equivalent to the linear partial differential equation of the first order in the same variables, /l+''|+4^=«> « and that two independent solutions of the latter were always two independent integral-functions of the former, and vice versa. Thus we may consider the linear partial equation (1) as taking the place of the above simultaneous system ; and when we speak of the equation (1) admitting of, or being invariant under a given 0^ — an expression which we shall immediately explain — we may also, if we choose, say that the simultaneous system admits of, or is invariant under the given 0^ The theory of integration of this section will be developed, therefore, for the linear partial differential equation (1), using that equation as the representative of the corresponding simultaneous system. 155. A O.^ in three variables has the general form ^f-€^4y+€ « where ^, »;, ^ are functions of the variables x, y, z. We say that the linear equation ^/■^l+^|+^l=« m — where X, Y, Z are, of course, certain functions of X, y, z — is invariant under, or admits of the 0^ Uf when, by means of the G^ Uf, each solution of (1) is trans- formed (compare Art. 58) into a solution of (1). Thus, if a)j{x, y, z), w^ix, y, z) be two independent solutions of (1), using the customary symbolic method for expressing the' fact that the transformation Uf is performed upon the function wi, the condition that (1) shall be invariant under Ufis, fr(o,i) = aK«2) i = l,2 (3) 198 ORDINARY DIFFERENTIAL EQUATIONS. This condition for the invariance of (1) can, of course, only be applied when the solutions (Oi, w^ are known; but we shall in the next article develop a condition which is practicable when w^ and w^ are unknown. 156. The expression U{Af)-A{Uf) has a definite meaning : it means, for the first term, put Af in place of f, in Uf; and, in the second term, put Uf in place off, in Af Thus it is seen UiAf)-Am.^l{x^J+Y^^+Z% If the differentiations here indicated are carried out, the terms involving differential coefficients of / of the second order will cancel out. For instance iX;-^ will occur in the first term with a positive sign, and in the fourth term with a negative sign, etc. Thus we find the noteworthy symbolic expression U(Af)-A{Uf) X^-dx^^dy^^dz dx dy dzjdx + Vd^+'>^ +^di -^^x~ ^d^'^dzjdz- INTEGRATION OF SIMULTANEOUS SYSTEMS. 199 But when it is remembered that and that similar expressions hold for U{Y), A{ri), etc., the above identity may be written (putting for brevity U(Af)-A(Uf) Now if w^, Mg be the (unknown) solutions of Af=0, we must have A(a)^) = A(w2) = 0, Hence also U(A(a,,))^U(Aico,))^0. Further, from (3), if Af=0 admits of the 0^ Uf, A(U(a>i))^A(f^i(o>„ o,,))^'^ .A(^,)+^.A{u,,), i=l. 2. oajj O032 But since Wj, w^ are solutions of Af= 0, this last expres- sion must be zero identically. Thus the whole expression U{Af)—A(Uf) becomes zero if wi is put in place of /, and (4) becomes (UX-Ai)^^+{Uy-^',)^+iUZ-AO^^O. *=1,2 (5) Also we know that X^+Y^+Z'^^0; (6) ?a> ay oz so that from (5) and (6) must follow the identities, UX-Ai _ UY-Ar,_ UZ-A^ ^, X ~ Y ~ Z '^ ' 200 ORDINARY DIFFERENTIAL EQUATIONS. Let the value of the ratios (7) be represented by \{x,y,z); then we may write UX-Ai=\.X, UY-An = \.Y, UZ-A^^X.Z. Hence from (4) cr(^/)-^(cr/).x(x|+F|+^D.x.4/....(8) This then is the condition that a linear partial diflFer- ential equation of the first order shall admit of a given Gj^: and it is clear that the condition may at once be extended to n variables (including n = 2). It is customary to write (8) in the brief form {U,A)^\.Af, (9) where the left-hand member of (9) is merely an abbrevia- tion for the left-hand member of (8). It is easy to see that the necessary condition (8) is also sufficient. For if to; is put in place of / in (8), we obtain ^(£^(0,0)^0, (10) since the other terms in (8) vanish identically. But (10) means that U(wi) is a solution of Af= ; that is, if (8) is a true equation, the solutions wi must admit of the trans- formation Uf — that is, the differential equation Af=0 itself must admit of Uf. Hence, the necessary condition (8) is also sufficient. 157. It is evident from (9) and (8) that every expres- sion of the form (A, A) or (U, U) is identically zero, and hence the condition (9) that the equation Af=0 shall admit of the 0^ p . Af, where p is an arbitrary multiplier, is satisfied. But this transformation p . Af, which tells us nothing new concerning the equation Af=0, and which has therefore no value in the problem of integra- tion, is said to be trivial with respect to Af=0. This accords with the definition of Art. 60 for trivial trans- formation in two variables. Such transformations are always to be disregarded in our investigations. INTEGRATION OF SIMULTANEOUS SYSTEMS. 201 158. We shall now for the moment write TJf equal to zero, and consider the equation Uf= as a second linear partial differential equation, and we shall show that if the condition (9) exists, that is, if {U,A)^\.Af, (9) then Uf= and Af= may be put into forms for which {U, A) = 0, and ultimately that these equations have one solution in common. When the condition (9) holds, and X is not zero, the two linear partial equations Uf= 0, Af= are said to form a complete system of two members. When, in (9), X = 0, the two equations are said to form a Jacobian system of two members. For the sake of symmetry we shall assume the two linear partial differential equations of the first order in the forms AJ=0, AJ=0, and shall merely assume that they fulfil a relation of which (9) is a particular case, that is, we shall assume that the relation {A^, A;) = Pj{x, y, z)AJ+p2{x, y, z)AJ (11) exists. If jOj = 0, the condition (11) is identical with (9). As far as the maintenance of the condition (11) is concerned, we shall see that a condition of the form (11) must still hold when the equations Aj^f=0, AJ'=0 are replaced by any equations which are consequences of these two, as AJ^\.Af +\. A J=0, A J^ fji,. A J+fi,. A J=0, (12) where Xi, /xk are arbitrary multipliers, whose determinant, however, must evidently not be zero. 202 ORDINARY DIFFERENTIAL EQUATIONS. Hence {AJ, AJ) = {\A.^ + \A^, Mi^i + ^a^j) + (Xi . ^iMl + X2 • ^2Ml)^l/+(\ ■ -ilM2 + ^2 • ^2/"2)^2/ -(Mi . ^lXl + M2 • ^2^i)^i/-(Mi • ^ A2 + M2 ■ ^2>^2)^2/ Since we know that (J.j, ^j) = {A^, A^ = ; and since (as is easily verified) (A^, Aj) = —{A■^, A^), while the four last terms are aflFected by coefficients which are functions of X, y, z, it is clear that (^j/, A^) is an expression which is linear in terms of A-^f and A J, that is, by means of (12), {A^f, A J') is linear in terms of A-^f and AJ^. Thus, as far as the relation (11) is concerned, it is certain that the equations A^f=0, AJ'=0 may be re- placed by any equations of the form AJ=0, A.J=0, as given in (12). Let us therefore take the two linear partial equations in the forms, 3:/-|-<^x(^.2/.-)|=0, Z,/.|-.,(a=,2/,.)|=0. (13) Here {A^f^, -^2/) must still be capable of being written as a linear expression in terms of A-^f and A^f; but when the operation indicated by {A-^, A^ is carried out, it will be found by (13) that the result is free of ^ and ^ : that is, in the expression (A. A) = -Tl ■ Af+ T2 ■ A J, when A^f and ^2/ ^^^ chosen in the form (13), we must have Tj = Tg = : or, (AJ,AJ)^0 (14) Hence, if two linear partial differential equations of the first order satisfy a condition of the form (11), they may always be chosen in such a manner as to satisfy the condition (14). INTEGRATION OF SIMULTANEOUS SYSTEMS. 203 159. It now remains for us to show that if two given linear partial differential equations of the first order satisfy a condition of the form (14), they must have a common solution. If u and V be the solutions of A.^f=0, it is known that the most general solution of A.^f=0 must be some function of u and v of the form Q(u, v). We now wish to determine Q in such manner that it shall also be a solution of A^f=(i. We have AID. (u, ■")) = 3^ • ^2(tt) + 3^ ■ 3^2(^) ; and by means of the relation (14), putting u and v respec- tively for / in MA^f) -MA^f) = 0, (14) it is easy to see that, since ^i('M') = A-J^v) = 0, A^(A^{u)) = A^(A^{v))^0. That is to say, A^itu) and A^iv) are solutions of Aj^f=0, and are therefore functions of u and v, say, A^iu) = (j> {u, v), A^iv) = \lr{u, v). Hence MQ.iu, v)) = 0(u, v)^-\-ylr{u, v)^. The condition, therefore, that fl(it, v), which is a solution of J^j/=0, shall also be a solution of A^f=0, takes the form {u,v)^+i.{n,v)^^ = (15) This is a linear partial diflerential equation of the first order in u and v ; and it is always satisfied by the integral function of the corresponding system du _ dv 204 ORDINARY DIFFERENTIAL EQUATIONS. If this integral function be W{u, v), then W is the common solution of A^f=0 and A^f=0, the existence of which was to be proved. Of course TT is a function of x,y,z; so that Tf(a;, y, 2:) = const. represents a family of surfaces in space. 160. We shall now return to our original equations of Art. 158, having proved that the existence of the condition (9), that the equation Af= shall admit of the 6?^ JJf, means that the equations (16) form a complete system — that is, that they have one solution in common. If W be the common solution of (16), at any point X, y, z on one of the surfaces Tr= const., two tangential directions are assigned to the point by means of Iff and Af; and the direction cosines of these tangential direc- tions are proportional respectively to ^, ;;, f and X, Y, Z, Art. 19. If a, /3, y be three quantities proportional to the direction cosines of a line perpendicular to the above two tangential directions at the point x, y, z, we have Xa+Y^+Zy = 0, ^a+»?/3+fy = 0; whence, a^Yt-nZ. ^==Zi-^X. y = Xn-iY. If now dx, dy, dz represent the differential coefficients of the variables x, y, z on the surface W= const., it follows that the relation {Y^-riZ)dx+(Zi-^X)dv+{Xri-^Y)dz = 0... (17) must be satisfied by the coordinates of all the points on those surfaces. INTEGRATION OF SIMULTANEOUS SYSTEMS. 205 In other words, the common integral surfaces of the complete system Af= 0, Uf= satisfy the total differential equation (17). A method for integrating equations of the form (17) has been given in Art. 100, Chapter VIII. If 'W{x, y, z) = const. •be the integral required, we know that W will be one of the solutions of the given invariant linear partial differential equation. 161. Considering one of the solutions of the linear partial differential equation of the first order in three variables which admits of a known 0, as having been obtained, we shall now show that the omer solution may be found by a mere quadrature. To this end let us suppose that W(x, y, z), the solution already found, actually contains the variable z — for, of course, it must contain one at least of the three vari- ables — and in place of x, y, z, let x, y, W be introduced as new variables. In these variables Af by Art. 35, will have the form or, since by hypothesis, A{W) = 0, Af.Ax.%^+Ay.^. Now eliminate z from Ax and Ay, and Af will have the form Af^a{x,y,W)^^+^{x,y,W)^. Analogously, we find for the transformed Uf, Uf^y{x,y,wf^+S{x,y,W)% 206 ORDINARY DIFFERENTIAL EQUATIONS. We see that in Af=0 no differential coefficient with respect to W occurs at all ; and JJf does not transform this variable ; hence, W plays the role of a mere constant, and X and y are the only variables. The problem has now been reduced to the integration of^the linear partial differential equation in two variables Af=0, which admits of the known Q^, in the same two variables, JJf. But as this partial differential equation is equivalent to the ordinary differential equation, Art. 16, ady — ^dx = 0, which admits of Uf, the solution can be found, by the methods of Chapter IV., by a mere quadrature, in the form Tr=f ajic.y, W)dy-^(x,y, Ja{x,y, W).8(x,y,W)-^(x,y, W)dx W).y(x.y,Wy The integration here is to be performed as if TF were a constant, and afterward the value of W as a function of x, y, z is to be introduced. We may sum up the results of this section as follows : If a linear particd differential equation of the first order in three variables admits of a known infini- tesimal transformation which is not trivial, itsintegration may he accomplished by the integration of an ordinary differential equation of the first order in two variables, together with one quadrature. Example. The linear partial diflFerential equation of the first order Af^{a?+f+yz)^+(^^+y-^-xz)^+{xz^-yz)^=0 admits of the O^ since the application of the criterion (9) gives in this case {Uf,Af)^Af INTEGRATION OF SIMULTANEOUS SYSTEMS. 207 Thus Uf=Q and Af=0 form a complete system with a solution which is the integral of the ordinary diflFereutial equation (17). The latter equation will be found to reduce itself in this case to the form xzdx+yzdy — {x^+y-)ds=Q, (21) when the substitutions X = x''+y'^-\-yz, $=x, etc., are made. By the method of Chapter VIII. we find at once as the intearral of (21), ,3 W{x, y, z) = 5^±? = const. ; z and it may be readily verified that W is really a solution of both Uf=Q and Af^O, since it is found that U{W) = A{W) = 0. Now we shall introduce x, y, and W as new variables, eliminating '■ by means of Hence and ^J-''-dx^y?)y Hence, the second solution of Af—Q is 7=1 = log {V^+y2- W . (x-y)}. its value- second so 'Jx'^ +y^(y-\-z-x) If, now, in place of W, its value — in terms of x, y, z — be put, there results finally for the second solution. Fslog- 162. A theory of integration of an invariant linear partial dilTerential equation in n variables — that is, of an invariant simultaneous system in n variables — analogous to the theory of this paragraph for the invariant linear partial equationin three variables,might now be developed. 208 ORDINARY DIFFERENTIAL EQUATIONS. But a discussion, both of that theory and of the method of integration to be employed when a linear partial equa- tion is invariant under Tno-re than one known G^, must be reserved for a later occasion. SECTION III. Second Method for Ordinary Differential Equations of the Second Order in Two Variables. 163. The theory of integration of the last section may be readily applied to ordinary diiFerential equations of the second order in two variables which admit of a known 0^ For, Art. 152, it was seen that the differential equation of the second order y"-w{x,y,y') = (1) is equivalent to the simultaneous systemin three variables, dx_dy_ dy' , . 1 ~y'-w{x,y,y') ^""^ which, in turn, is equivalent to the linear partial diflfer- ential equation of the first order in three variables, ^/-|+2''|+'«(^'2''2/')|=0 (3) Thus, if the differential equation (1) admits of a known twice-extended 0^ U"f, the partial differential equation (3) must admit of the once-extended Cj, 'dy'-'dy'. that is, we must have (U',A)^X.Af (4) SECOND ORDER IN TWO VARIABLES. 209 Thus the condition (9) of the foregoing section is satisfied, and the partial differential equation (3) may be integrated by the methods of that section. That is to say, Art. 100, if an ordinary differential equation of the second order in two variables, y"-w{x,y,y')=0, admits of a known G^, the differential equation of the second order may be completely integrated by the inte- gration of an ordinary differential equation of the first order in two variables, and a quadrature. Example. Given This equation may be written y"-{H)='-' (^> and hence (3) has the form It may be at once verified that (5) admits of the Oi to which corresponds the once-extended (?i ^j-'^-dx y?>y" so that the condition (9), Sec. 2, is satisfied, and the equations Af=0, Uy=0 (6) form a complete system. Tlie common solution of the equations (6) must be the integral- function of the total equation corresponding to (17), Sec. 2, i/y'dx-(2y-xi/')dy+xydy' = (7) By Art. 99, or Art. 100, the integral-function of (7) is found to be W(x,y,y')=f-xyy'. (8) P.O. o 210 ORDINARY DIFFERENTIAL EQUATIONS. Now introduce x, y, and W — the common solution of the equations (6) — as new variables ; thus, Eliminating y' from Afhy means of (8), we find ^ "dx oey "dx Now Af=0, a linear partial equation in the variables x and y, admits of ty in the same variables : hence the second solution of Af=0 is found by a quadrature in the form F=^- = — . f - W yy" Thus, eliminating y" between W=y^-xyy' = ci and V=^ = C2; we find or as it may be written ma^+ny^ = l, (m, w consts.) which is the form of the complete integral of (5). Thus we see that (5) represents the oo^ conic sections whose axes coincide with the axes of coordinates ; and it is clear, geometrically, that this family of co* curves is invariant under the Cj of afSne transformations 164. The simultaneous systems given in the following Examples are simple ; and, for the most part, they may be integrated by the methods of both Sec. I. and Sec. II. Examples (1) to (4) are, however, intended to illustrate Arts. 148, 149; Examples (5) to (14), Arts. 150-153; while the remaining Examples are intended to be treated by the method of Sec. II., after it has been verified in each case. Art. 156, that the given simultaneous system is invariant under the accompanying G^ Examples illus- trating Sec. III. may be found at the end of Chapter IX. EXAMPLES. 211 EXAMPLES. (1) dx _ dy dz x^—y^ — ^2 2xy 2xz' -g, Idx _ mdy _ ndz '^ mn(j)-zynl{z~x)~lm{x-y") /ON Idx mdy _ ndz {m - n)yz ~(n-l)zx~ {l-m)xy' (4) dx _ dy _ dz_ ' x{y-z) y{z-x) z{x-y)' C5^ dx_^_dy_^_dz__ ^ ' -\ 3y + 4z 2y + bz' ^■'z+2y-2a? z^-x-by z' its dx _ dy dt (8)g+5:.+y=e«; J + 3y-x = ««. ,(,\ dx dy _ (13) J-3^-4y + 3 = 0, g + ;r-8y + 5 = 0. ns) dx dy ^ dz _ j7yr_2^^§/; :;;— y — 2+2 '2,{y-x + z) x—y — z' ■^ 'dx 'dz 212 ORDINARY DIFFERENTIAL EQUATIONS. , , dx _ dy dz . ^.Jo_'df 2_3/_ ^ ' xz + <^-''~-z{\+x) 0(e*-=-2)' ^ \+x'dx \+x^z (17)J^ = ^L= dz jy^(^+ )|/+(^+ )^+22|f. ^ ' x-k-y x+y -{x+y + 2z)' ■' ^ "'^x ^ ^'3y Oz (18) J^ = Jy-^^^., Uf^x'^+y% ^ ' xz-y yz-x \-z^ ■' OX ^Qy (19)_^ = % = ^; Uf^M. z-'2x xz+yz + 2x-z z ' ■' ay (20) ^=^= — dz cr/^|:+|:^|: y-z y — z {x-y){x-z) ■' ox ay oz (21) Verify that the linear partial differential equation correspond- ing to the simultaneous system, dx _ dy _ dz a^x + biy + CiZ+di~ a2X + b2y + c^ + d2~agX + b^ + C3i + d3 admits of a O^ of the form, Uf^^^+a)^+(y + /3)^+(z+y)'^^, where a, fS, y are certain constants, and that therefore the above simultaneous system may usually be integrated by the method of Sec. II. (22) The method of Sec. II. fails for the preceding example only in the case of TJf being trivial. For what values of the con- stants a,, 6i, ... , ^3 is Uf trivial ? What are then the integrals of the simultaneous system ? (23) Verify that the linear partial differential equation correspond- ing to the simultaneous system, dx _dy _ dz admits of the G^ ^J~''?>x^y-dy^''dz' if X, T, Z are homogeneous functions of x, y, z; and that therefore, when C^is not trivial, the integration of the above system may be reduced by the method of Sec. II. EXAMPLES. 213 (24) Verify that the linear partial differential equations correspond- ing to the simultaneous systems in Ex. (16) and (18) admit, respectively, of the G^s, ,„ ox ' oy oz . ^^" RT^f Uf^ (^+y2)g+2^g-y(l -.»)|. (25) Verify that the linear partial differential equation, ^/■i-|-(^-x')'*{|)I=o. admits of the O^, ANSWERS. CHAPTER I. (i)y=f. (5) (H-^)y+5^ = tan-ix (7) y=xy+y'-y'3_ (9; ^y'-2^ + 2y=0. (11) :r3y' + (y_^-)2^o (13) f' = 1y'-Qy. (15)y2(l+y2)=r2 (17) l+y2_(y_^y)2 = o. (19) xY'-xy'+y=Q. (2) y=xy'+s/T+f'. (4) yy^ + 2:!ry=y. (6) ^' + !/=j/21og^ (8) ^Y2 = l+y2. (10) y"+m2y = 0. (12) xY-xy'==Zy. (u)y"-2y' +/=««. (16)ry'2 = (l+y2)3. (18) xyy"+xy'^-y!/=0. (1) »2J;2_y + cy + 2=0. CHAPTEE II. (2) (H-.x.)(n.y)=c. (4) 3(a;2-/) + 2(.j;3_^3)^g_ (5) coay = ccosar. (6) iog[Cy+v/r+p)s/r+p-j=^_+c. vl+,r2 (7) Bin2.j;+8in2^=c. (8)y=c,.»-; y2=c2. (9) ^2+y2=c,2; tan-i2!-iog2=c2. ANSWERS. 215 (10) x^-iry^=e^; tan i^-tan-iz = const., or taking the tangent of X both aides, — — ^=Co. ' x+yz '■ {^\) a^-y-'^e^; y^-z'^c^. {\^) x = c,; y^+z^ = c^. CHAPTEE III. N.B. — Only such invariant points and lines as are within a finite distance from the origin will be taken into consideration. (1) No invariant point. An Invariant is Q,(j/). (2) An Invariant is Q,{x). (3) All points on the y-axia invariant. An Invariant is 12 (y). (4) The origin is an invariant point. An Invariant is 12 ("j. (5) The origin is an invariant point. An Invariant is J2fi j. (6) The origin is an invariant point. An Invariant is Sl(xy). (7) The origin is an invariant point. An Invariant is I2(x*+y^). (8) All points on the y-axis are invariant. An Invariant is fli " I. (9) All points on the a^-axis are invariant. An Invariant is 12 ( - J. CHAPTER IV. (1) a?-&x'hf-&xy^+f = c. (2) ifi-y'^=c.f. (3) x+ye'=c. (4) coa(mx+ny) + Bm(nx+my)=c. (5) »/l+^2+^2 + tan-i- = c. (6) e'(.v'+y^) = c. ^ _ /i (7) y = c.e «. (8) y = ee "^ ". (9) x2 = c2 + 2cy. (10) log{x^+y^) = 2 tan"*^ + c. (11) x^^ce'""'. (12) (2/ + xf(2/ + 2xy=c. (13) sin-i^ = loga; + c. (14) x^y*=yV + c. (15) x^+y'^cxy. (16) x'-xy+y'^+x-y^c. 216 ORDINAR T DIFFERENTIAL EQ UA TIONS. (17) {y-x+\f{y+x-\f=c. (18) ^^Vtan-i^ = c. (19) xy^^^cix+iy). (20) y = cx. (21)y=ce«. {2'i.) xy- — =\ogcy''. ex (23) a^=c. (24) xy + \ogsm{xy) = \og 3 (25) 2. = c^a^+J^-l (26) y=-^^+c«^. (27) ;p=tan-'y-l+c.e-'""'* (28) i-2=^+J + ce^. (29) - = cVr3p_i. (30)- = loga;+l + ai7. (31) Admits of Cy=ar^+y^. Ana. log^+|'=c. (32) yVr+^=log ^' ~ +c. (33) 3^(seca:+tanj7)=a;+c. (34)y={cVr^-o}-'. (35) y^{c^Jr3?+\Y^. (36) y={cj;+log.r + l }-'. ^3^. ^tan£+aeca: -gg. 5^,2 = 2 sin j;+4cosa: + ce-^. (39) Admits of C^/-£ |+y| Ans. |-^ = c. (40)y"=c.^. (41) y =0.6*. (42) y2 = 4-l+V^) (7)y'==c.^. (8) Isothermal. x^+y^ + a/+\=0. (9) The system is self -orthogonal. (10) y= f;^N+ const. (U) 2/= -i-Jni+c. (13) ? = 8in2e-|-c. (14) ^=-|-^^-^ const. (15) g=|cos-'|- ^^'~^' |-t-const. CHAPTER VI. (1) (j/-'2x+c){y-Zx + c) = 0. (2) y = c.^, y^c.e-"- (3) (xy + c){xh/ + c)=0. (4) {x^-'iy + c){{x+y-\y + c)=0. (5) {y+c){y+x^-^c){xy-^cy+V)=0. (6) y28iii2a;+2cy-t-c2=0. (7) Admits of Cr/^(l-H^)g+y^. General integral, i+^cx^+C4e-'^. 6) y = e'(ci + c^+ c^ + c^x^). 7) y = e^»{cj sin (a" -V)x+ c^ cos (a'' - l>')x\. 8) ye* = Cj . e^ + Cj sin W2 + c^ cos W2. 9) y=Ci. e'^+ C2e~'^+ Cg sin 2x + C4 cos 2a;. 0) y=Cie-*+(c2+Cjr)e^. 1 ) y = (cj + Cjj;) cos »ur + (C3 + 64^) cos 2a;. 19r-l-7 2) y=(ci + C2a;+C3a;2)e'+C4. (13) y=Cie^+C2e''+i^^. 4) y = CiSina;+C2C0sa;+(c3 + C4a;)e'+l. 5) y={ci + c^ + ^)e' + c, 6) y=CiSianx + C2 00snx + 3- l+a;+a;2 2 222 ORDINA R Y DIFFERENTIA L EQ UA TIONS. (19) 2/=[ci - ^j sin 2a;+ (cj -^jcos 2^ + |. (20) y = c,e-^ + (c^ - ^) e- cos ^ + (cj + g) e- ain ^. (21) y = c,^ + ^ (22) 3/ = c, (^ + a)2 + c,(x + af + ?^±^. (23) y=^(cisinlog:r+C2Cosloga;+log;i;). (24)y=(2^-l)|ci + C2(2:r-l) 2 +63(2^-]) ^ }. (25) A particular integral-function is x. General integral : y= -|(1 -:r2)' + Ci{^sin-'a; + (I -.r2)*} + c2.r. (26) A particular integral-function is e'. General integral : y = e'{\e 2 {\xe » ofo-f-cJote-l-Ca}. (27) A particular integral-function is -. X General integral : y = -^-h ' ^ ■ (28) A particular integral-function is -, X General integral : y = c-^^ + c^ogx-\- X x^ 4" (29) A particular integral-function is e''° '*. General integral : .!/ = Cie"""''-l-C2e'=°''~'*. (30) A particular integral-function is e'. General integral : y = Cie'-(-C2^-(.j:^-t-ar-(-l). ANSWERS. 223 CHAPTEE XII. (l)y = CiZ; a^+f+z''=Ci.z. (2) Px+nv^ + nh=c^; lV + mY + nV = C2. (3) Ix^ + mf + nz^ = c, ; IV + m.y + m V = c^ (4) x+i/+z = Ci ; xyz = c^. {b) y=- 2cie- ' + c^-'" ; z = 0^6'' +0^6-'". (6) .= .,.-4 + 2c^-3 + g + ^. y=_,,,-4_,^-3_^ + g. 2 71 (7) 2j;=(2c2-Ci-C20e"'; y=(c, + C20e"'. (8) ^=(c, + CjOe-*--+— ; y= -{c, + c, + c,t)e-*'+'-^+^^. (9) x=2c,«--c^-r. + g+g; y = ..«-4.+,^e-" + ^+^. (10) x={c^Bint + CiCoat)e-^ + ^-^; ^ = {(c2-Ci)sin<-(c2 + Ci)oos«}e-«'- — + — . _, , , 5, , 24e' 17< 56 3^= -Cie '+4C2« «+-^ "T+y (12) ^=CiSinK<+C2C0SK< ; y = c^ + c^t — x. (13) ^=4Cie»+4c2e-'»+C3e''*^+C4e-''^' + }, (14) y = (ci + Cisa')«'+3c3e «--; z = 2(3c2-Ci-C2a:)e*-Cge"^-J. (15) y + 2z = Ci; 2a;+y-21og(a;-y-2+l) = C2. (16) e^ + e*=Ci ; a;+3^ -log 2=02- (17) a;-y=Ci; ^2+^2+05^ = 02. (18) 3/+.r2=c, ; x+yz=C2. (19) x=c,2-'' + |; y = C2e'-Ci2-^-g. 224 ORDINARY DIFFERENTIAL EQUATIONS. (20) :t-y=c, ; (22) The (?i is trivial for with the rest of the constants zero. The integrals in this case are x + a . y + fi . ?,=const., - — '-- = const. y+P z+y .jiJtou.M\iVERSITYLIBRA(<. OCT ij 199] AlAThEMATJCSUSHABV