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Cornell University 
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 There are no known copyright restrictions in 
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QJarnell Untttwaitg ffiihrarg 
 
 ..Xo-bri y..e..o..ir.y..Xa.n.n er.. 
 
 IIATHEbAUUS 
 
EXAMPLES 
 
 DIFFERENTIAL EQUATIONS 
 
 RULES FOR THEIR SOLUTION. 
 
 BY 
 
 GEORGE A. OSBORNE, S.B. 
 
 Pbofesbob of Uatheuaticb in the Massachusetts Institute 
 or Techmoloqy. 
 
 oMrto 
 
 BOSTON, U.S.A.: 
 
 PUBLISHED BY GINN & COMPANY. 
 
 1889. 
 

 3-;. 
 
 Entered, according to Act of CongreeBf in the year 1886, by 
 
 GEORGE A. OSBORNE, 
 in the Office of the Librarian of CongreSB, at WaablngtOD. 
 
 J. S. Gushing & Co., Pbintbbs, Bobton. 
 
PREFACE. 
 
 THIS work has been prepared to meet a want felt 
 by the author in a practical course on the subject, 
 arranged for advanced students in Physics. It is in- 
 tended to be used in connection with lectures on the 
 theory of Differential Equations and the derivation of 
 the methods of solution. 
 
 Many of the examples have been collected from standard 
 treatises, but a considerable number have been prepared 
 by the author to illustrate special difficulties, or to pro- 
 vide exercises corresponding more nearly with the abilities 
 of average students. With few exceptions they have aU 
 been tested by use in the class-room. 
 
 G. A. Osborne. 
 
 Boston, Feb. 1, 1886. 
 
COI^TEITTS. 
 
 CHAPTER I. 
 
 DEFINITIONS. — DERIVATION OF THE DIFFERENTIAL EQUATION FROM THE 
 COMPLETE PRIMITIVE. 
 
 PAGE. 
 
 Definitions 1 
 
 DeriTation of difEerential equations of the first order 1 
 
 Derivation of differential equations of higher orders 2 
 
 SOLUTION OF DIFFERENTIAL EQUATIONS. 
 CHAPTER II. 
 
 DIFFERENTIAL EQUATIONS OF FIRST ORDER AND DEGREE BETWEEN TWO 
 
 VARIABLES. 
 
 Form, XYdx-^-X'Yidy = (i 4 
 
 Homogeneous equations 5 
 
 Form, {ax + 5y + c) rfx + (a'x -\-Vy-{-c')dy = Q 5 
 
 Linear form, -^ + Py — Q 6 
 
 dx 
 
 Form, ^■¥Py=Qr 7 
 
 dx 
 
 CHAPTER III. 
 
 EXACT DIFFERENTIAL EQUATIONS. — INTEGRATING FACTORS. 
 
 Solution of exact differential equations 8 
 
VI CONTENTS. 
 
 Solution by means of an integrating factor in the following cases : 
 
 Homogeneous equations 9 
 
 Form, fi(xi/)ydx+f,(xy)xdi/ = , 10 
 
 dM_dN dN dM 
 
 TIT, du dx , ^ dx dii , , . -„ 
 
 Wlien -1-^^^ = <t>{x), or —J-=,p(y) 10 
 
 CHAPTER IV. 
 
 DIFFERENTIAL EQUATIONS OF FIRST ORDER AND DEGREE WITH THREE 
 VARIABLES. 
 
 Condition for a single primitive, and method of solution 12 
 
 CHAPTER V. 
 
 DIFFERENTIAL EQUATIONS OF THE FIRST ORDER OF HIGHER DEGREES. 
 
 Equations which can be solved with respect to p 14 
 
 Equations which can be solved with respect to y 14 
 
 Equations which can be solved with respect to x 15 
 
 Homogeneous equations 16 
 
 Clairaut's form, y =px +/(p) 16 
 
 CHAPTER VI. 
 
 SINGULAR SOLUTIONS. 
 
 Method of deriving the singular solution either from the complete 
 primitive or from the differential equation 17 
 
 DIFFERENTIAL EQUATIONS OF HIGHER ORDERS. 
 CHAPTER VII. 
 
 LINEAR DIFFERENTIAL EQUATIONS. 
 
 Linear equations with constant coefficients and second member zero . . 19 
 Linear equations with constant coefficients and second member not zero, 22 
 A special form of linear equations with variable coefficients 24 
 
CONTENTS. VU 
 
 CHAPTER VIII. 
 
 SPECIAI. FORMS OF DIFFERENTIAL EQUATIONS OF HIGHER ORDERS. 
 
 PAOB. 
 
 Form, ^=X 25 
 
 Form, ^ = r 25 
 
 Equations not containing y directly 26 
 
 Equations not containing x directly 27 
 
 CHAPTEE EX. 
 
 SIMULTANEOUS DIFFERENTIAL EQUATIONS. 
 
 Simultaneous equations of first order 28 
 
 Simultaneous equations of higher orders 30 
 
 CHAPTEE X. 
 Geometrical applications S2 
 
 Answers to Examples 35 
 
EXAMPLES OF DIFFERENTIAL EQUATIONS. 
 
 <»»;o 
 
 CHAPTER I. 
 
 DEFINITIONS. DEBIVATION OF THE DIFFERENTIAL EQUA- 
 TION FKOM THE COMPLETE PRIMITIVE. 
 
 1. A differential equation is an equation containing differen- 
 tials or differential coefficients. 
 
 The solution of a differential equation is the determination of 
 another equation free from differentials or differential coeflB- 
 cients, from which the former may be derived by differentiation. 
 
 The order of a differential equation is that of the highest 
 differential coefficient it contains ; and its degree is that of the 
 highest power to which this highest differential coefficient is 
 raised, after the equation is freed from fractions and radicals. 
 
 The solution of a differential equation requires one or more 
 integrations, each of which introduces an arbitrary constant. 
 The most general solution of a differential equation of the nth 
 order contains n arbitrary constants, whatever may be its de- 
 gree. This general solution is called the complete primitive of 
 the given differential equation. 
 
 2. To derive a differential equation of the first order from its 
 complete primitive. 
 
 Differentiate the primitive ; and if the arbitrary constant has 
 disappeared, the result is the required differential equation. If 
 not, the elimination of this constant between the two equations 
 will give the differential equation. 
 
a DEEIVATION OF THE DIFFERENTIAL EQUATION. 
 
 3. Form the differential equations of the first order of which 
 the following are the complete primitives, c being the arbitrary 
 constant : 
 
 1. 
 
 log {xy) + a; = 2/ + c. 
 
 2. 
 
 {l+a?){l+f)=c3?. 
 
 3. 
 
 cos y= coos X. 
 
 4. 
 
 y = ce-"""'' + tan->a; — 1 
 
 5. 
 
 y= (ca; + loga;4-l)"^ 
 
 6. 
 
 y = cx + c — c'. 
 
 7. 
 
 {y + cy = 4cax. 
 
 8. 
 
 y^ sin^a; + 2 cj/ + c^ = 0. 
 
 9. 
 
 e''' + 2cxe>'+<f = 0. 
 
 4. To derive a differential equation of the second order from 
 its complete primitive. 
 
 Differentiate the primitive twice successively, and eliminate, 
 if necessary, the two arbitrary constants between the three 
 equations. 
 
 5. Form the differential equations of the second order of 
 which the following are the complete primitives, Cj and Cj being 
 the arbitrary constants : 
 
 1. y = CiC0B{ax+C2). 
 
 2. ?/ = Cie'" + C2e-". 
 
 3. y=(ci + C2x)e". 
 
 4. y = Cioe' + ~- 
 
 F . , , cosaa; 
 
 5. y = Cismnx + C2Coanx + — ■• 
 
DERIVATION OF THE DITFEEENTIAIj EQUATION. 8 
 
 6. The preceding process may be extended to the derivation 
 of equations of higher orders from their primitives. 
 
 7. Form the differential equations of the third order of which 
 the following are the complete primitives : 
 
 1. 2/ = Cie='= + C2e-=' + C3e'. 
 
 2. i/e* = Cie^' + C2sina;V2 +C3C0sa;V2. 
 
 3. 2/ = rci + C2a; + yy + c3. 
 
 Form the differential equations of the fourth order of which 
 the following are the complete primitives : 
 
 4- y= {ci + Ci^ + C3a^)e' + c^. 
 
 6. a? + a*y = c^e" + CiC"" + Cssinoa; + Cicosaa;. 
 
CHAPTER II. 
 
 DIFFEBENTIAL EQUATIONS OP THE FIRST ORDER AND 
 FIRST DEGREE BETWEEN TWO VARIABLES. 
 
 General Form, Mdx + N'dy = 0, 
 where M, N, are each functions of x and y. 
 
 8. Form, XYdx + X'Y'dy = Q, 
 
 where X, X', are functions of x alone, and Y, Y', functions of 
 y alone. 
 
 Divide so as to separate the variables, and integrate each 
 part separately. 
 
 9. Solve the. following equations : 
 
 1. {l-i-x)ydx+{l-y)xdy = 0. 
 
 2. (^x'-yx')^ + f + xf = Q. 
 
 ax 
 
 3. 
 
 dx (1 + x')xy 
 
 ( dy 
 
 \ dx 
 
 4. a[x^^ + 2y] = xy^- 
 
 5. {l+f)dx={y+^\+f){\+a?y^dy. 
 
 6. sin a; cos ydx= cos a; sin 2/(^3/. 
 
 7. sec^xtan?/tZa; + sec^?/tana;d2/ = 0. 
 
 8. sec^K tan 2/^2/ + sec^2' tan a;da; = Q. 
 
 Q dy ^ l+2/ + y' ^Q_ 
 dx \ +x + x' 
 
HOMOGENEOUS EQUATIONS. 5 
 
 10. Homogeneous equations. 
 
 Substitute y=:vx; in the resulting equation between v and x, 
 the variables can be separated. (See Art. 8.) 
 
 U. Solve the following equations : 
 
 1. {y — x)dy-irydx=(i. 
 
 2. {2^\/xy — x)dy + ydx = 0. 
 
 3. f + a?^ = xy^. 
 
 dx dx 
 
 4. x-^ = y + '\/x' + y\ 
 
 dx 
 
 6. xcoas-.-K^^y cosi- — X. 
 X dx X 
 
 6. {%y + 10x)dx+{f)y + lx)dy = Q. 
 
 7. (x + y)-^ = y — x. 
 
 8. xcos^ (ydx + xdy) = y sin^ (xdy — y dx) . 
 
 X X 
 
 9. x + y-^ = m.y. 
 
 dx 
 
 (1), m<2; (2), m = 2; (3), m>2. 
 
 , dy 
 
 10. [ (a;^ — y^) sin a + 2 xy cosa — y wx' + y'J "^ 
 
 = 2 ojz/ sin a — (a^ — y") cosa + x -y/a^ + y'. 
 
 12. Form, 
 
 {ax + by-{-c)dx+ (a'x + h'y + c')dy = 0. 
 
 Substitute x = x' + a, y = y'+Pi 
 
 and determine the constants a, /3, so that the new equation be- 
 tween x' and y' may be homogeneous. (See Art. 10.) 
 
6 LINEAR EQUATIONS OF FIRST ORDER. 
 
 This method fails when —-=-—• In this case put ax + by = z, 
 a' o' 
 
 and obtain a new equation between x and z or between y and « ; 
 the variables can then be separated. 
 
 13. Solve the following equations : 
 
 1. {3y-7x + 7)dx+(7y-Sx + 3)dy = 0. 
 
 2. (4a; + 22/-l)^ + 2a; + 2/ + l = 0. 
 
 Ox 
 
 3 %^ 7y+x + 2 
 dx 3x + 5y + 6 
 
 4. (2y + x+l)dx={2x + 4:y + 3)dy. 
 
 5. 2x-y + l + {x+y-2)^ = 0. 
 
 ax 
 
 14. Linear Form, -/ + ^2/ = Q, 
 
 dx 
 
 where P, Q, are independent of y. 
 
 Solution, y = e~-'''^ijQe'^'^ dx + ch 
 
 15. Solve the following equations : 
 
 1. x— — ay = x + l. 
 
 dx 
 
 2. x{l — !ii?)dy-{-{2aP — l)ydx=a!iPdx. 
 idy 
 
 da; 
 
 4. — ^ + «cosx = -sin2a;. 
 dx " 2 
 
 5. (^l+y^)dx= (ta.n~^y — x)dy. 
 
EXTENSION OP LINEAR FOKM. 7 
 
 \ dxj 
 7. {l+r')dy + fxy--\dx=Q. 
 
 dx dx dx 
 
 where ^ is a function of x alone. 
 
 16. Form, ^ + Py=Qy'', 
 
 dx 
 
 where P, Q, are independent of y. 
 
 Divide by y", and substitute z = y~'+^. The new equation 
 between z and x will be linear. (See Art. 14.) 
 
 17. Solve the following equations : 
 
 1. {1 —x')-^ — xy=<ixy^. 
 
 2. 3f^-af = x + l. 
 
 dx 
 
 3. ^=2xif{cuicPf-l). 
 
 4. ^(x'f + xy)=l. 
 dx 
 
 5. — + yco8a; = ?/"sin2a;. 
 ox 
 
 6. (y\ogx—l)ydx = xdy. 
 
 7. a3?y"dy + ydx = 2xdy. 
 
 8. 2/ — cosa;— =2/'cosa;(l — sinx). 
 
 dx 
 
 9. y-^ + by' = acoax. 
 "dx " 
 
CHAPTER III. 
 
 EXACT DIFFERENTIAL EQt7ATIONS AND INTEGKATING 
 FACTORS. 
 
 18. Mdx + Ndy is an exact differential when 
 
 dy dx 
 The solution of 
 
 Mdx+Ndy =0, in this case is 
 
 (1) 
 
 or 
 
 ClUdx + ('(n- — CMdx\dy = ( 
 CNdy + ('(m- — CNdy\x = c 
 
 In integrating with respect to x, y is regarded as constant, 
 and in integrating with respect to y, x is regarded as constant. 
 
 19. Solve the following equations after applying the condi- 
 tion (1) for an exact differential: 
 
 1. {3? + 3xy^)dx+(f + 3afy)dy=0. 
 
 2. (a^-ixy—2y^)dx+ {y'' — Axy -2a?)dy =0. 
 \+^]dx-^-ldy = 0. 
 
 4. 2^-+a-3A,,=o. 
 
 r W y J 
 
EXACT DtPPBEENTIAI, EQUATIONS. 
 
 6. '^ +(l ^l—]^ = 0. 
 
 Vs^Tf \ ^3? + f) y 
 
 7. (x+ /^ - \dx4-(y Vy = 0. 
 
 8. (l+e'')da; + e»['l--^d2/ = 0. 
 
 9. e(p? + y^-\-1x)dx+'iyedy = ^. 
 
 10. (mdx -\- ndy^^vaimx -\- ny^ = (ndx+mdy) cos (nx+my). 
 
 jj_ xdx + ydy ydx — xdy ^ 
 
 Vl + ar* + 2/2 a;2 ^ 2^2 
 
 12. «°'^y-ay^-^dx ^^,.^^ 
 
 by^ — gar'' 
 (l),g>0; (2), 5r<0 and =-fc; (3), gr = 0; 
 (4), a = 0; (5), 6 = 0. 
 
 20. When Jfcia; + iVdy is not an exact differential, it may 
 sometimes be made exact by multiplying by a factor, called an 
 integrating factor. The following are some of the cases where 
 this is possible. 
 
 21. When Mdx + Ndy is homogeneous, is an in- 
 
 Mx + Ny 
 
 tegrating factor. This fails when Mx + Ny=Q, but in that 
 case the solution is j/ = ex. 
 
 22. Solve the following equations by means of an integrating 
 factor : 
 
 1. (x' + 2xy — y')dx={a^—2xy — y^)dy. 
 
 a dx 
 
 ■ + 
 X 
 
 y \y ^J ' 
 
10 INTEGRATING PACTOKS. 
 
 3. {o?y'^-'rX]f)dx— {3?y + 3?y^)dy = 0. 
 
 4. a?dx+ {^a?y + if)dy = 0. 
 
 5. {xy/af -'f y'' — a?) dy + {xy — y Va^ + y^) dx — 0. 
 (See Art. 11 for other examples.) 
 
 23. Form, fi{xy)ydx+fi{xy)xdy = Q. 
 1 
 
 is an integrating factor. This fails when 
 
 Mx — Ny 
 
 Mx — Ny^ 0, 
 
 but in that case the solution is xy = c. 
 
 Another method of solving is to put xy — v, and obtain an 
 equation between x and v or between y and v. The variables 
 can then be separated. 
 
 24. Solve the following equations by means of an integrating 
 factor : 
 
 1. {1 -\-xy)ydx + {l —xy)xdy = 0. 
 
 2. {a?y^-\-xy)ydx+ {3?y^ — l)xdy = 0. 
 
 3. {oi?f+l){xdyJrydx) + {x'y^ + xy){ydx — xdy) = 0. 
 
 4. {y/xy — l)xdy — {-s/xy + l)ydx=0. 
 5- {y + yyfxy)dx+ {x-irxy/xy)dy = 0. 
 
 6. e'''{a?y'^ + xy){xdy + ydx) +ydx — xdy = 0. 
 
 7. xy[_l +cot{xy)^{xdy +ydx) +xdy — ydx = 0. 
 
 dM_d_N 
 
 25. When -^ = <!>{«'), 
 
 then e-' "^ "" is an integrating factor. 
 
INTEGBATING FACTORS. 11 
 
 dN dM 
 
 ^ , dx dy . . 
 
 Or, when ^-2- = ,/,(2/) , 
 
 then e'^ " " is an integrating factor. 
 
 26. Solve the following equations by means of an integrat- 
 ing factor : 
 
 1. (x'+y^+2x)dx + 2ydy = 0. 
 
 2. (3x'-y^)^=2xy. 
 
 dx 
 
 3. 
 
 dy^o^jj-f 
 dx 2xy 
 
 4. 1(1 - y)^l - x' - xy^dx + [l - x' - x^/1 - x'^dy = 0. 
 
 5 . (cos X + 2y secAf sec^ 2x) da; + (tan 2a; sec y — sin a; tan y)dy=Q. 
 
 i 
 
 6. sin(3a;— 22/)(2da; — dy) +sin{x — 2y)dy = 0. 
 
 7. The Linear Equation 
 
 dy_^ 
 dx 
 
 where P and Q are independent of y. 
 
 dx 
 
CHAPTER IV. 
 
 DIFFERENTIAL EQUATIONS OF THE FIRST ORDER AND 
 DEGREE CONTAINING THREE VARIABLES. 
 
 General form, Pdx + Qdy + Bdz = 0, 
 where P, Q, B, are each functions of x, y, z. 
 
 27. If the variables can be separated, solve by integrating 
 the parts separately. 
 
 The equation is derivable from a single primitive only when 
 the following condition is satisfied : 
 
 pfdQ_dR,_^QfdR_dP\^j^fdP_dQ\^^_ 
 \dz dy J \ dx dz J \dy dxj 
 
 The solution may then be effected by first solving the equa- 
 tion with one of the parts Pdx, Qdy, Bdz, omitted, regarding 
 X, y, z, respectively, constant. 
 
 Omitting Bdz, for example, we solve Pdx + Qdy = 0, re- 
 garding z constant, and introducing instead of an arbitrary 
 constant of integration, Z, an undetermined function ot'z, which 
 must be subsequently determined so that this primitive may 
 satisfy the given differential equation. The equation of condi- 
 tion for determining Z will ultimately involve only Z and z. 
 
 28. Solve the following equations after applying the condi- 
 tion (1) for a single primitive : 
 
 1. dx dy ^ dz ^ ^^ 
 x — a y — b z — c 
 
 2. {x — 3y — z)dx + {2y — 3x)dy + {z — x)dz = 0. 
 
EQUATIONS CONTAINING THREE VARIABLES. 13 
 
 3. {y + z)dx + (z + x)dy + (x + y)dz = 0. 
 
 4. yzdx + zxdy -i'Xydz = 0. 
 6. {y + z)dx + dy-\-dz=0. 
 
 6. ay^z^dx+bz^x'dy+ca^y''dz=0. 
 
 7. zydx = zxdy + j/^ dz. 
 
 8. {ydx + xdy){a + z) = a^dz. 
 
 9. (2/ + a)^dx + 2d?/ = (2/ + a)d«. 
 
 10. (y^ + 2/2) da; + (as? + z') dy + (j/'' — a;?/) dz = 0. 
 
 11. {2a? +2xy + 2xz^+\)dx-\-dy + '2zdz = Q. 
 
 12. (ar'?/ — y'—y^z)dx+ (xy'—a?z—x'')dy+ {xy^+a?y)dz = 0. 
 
CHAPTER V. 
 
 DIFFEKENTIAL EQUATIONS OF THE FIRST ORDER, OF A 
 DEGREE ABOVE THE FIRST. 
 
 In what follows, p denotes — • 
 
 ax 
 
 29. Wlien the equation can he solved with respect to p. 
 
 The different values of p constitute so many differential equa- 
 tions of the first degree, which must be solved separately, using 
 the same character for the arbitrary constant in all. 
 
 If the terms of each of these separate primitives be transposed 
 to the first member, the product of these first members placed 
 equal to zero will be the complete primitive. 
 
 30. Solve the following equations : 
 
 1. 2)2-5j9 + 6 = 0. 
 
 2. x'p' - a= = 0. 
 
 3. V-o = 0. 
 
 4. xp^ = 1 — x. 
 
 5. x'p'^ + Zxyp + ^y^ = 0. 
 
 6. p{p + y)-=x{x + y). 
 
 7. p" + 2 ap^ — y-p^ — 2 xy'^p = 0. 
 
 8. p^-{3? + xy + f)p^ + {x'y + :^f + xf)p - x^f = 0. 
 
 9. p^ + 2^2/ cota; = y^. 
 
 31. When the equation can be solved with respect to y. 
 Differentiate, regarding p variable as well as x and j/, and 
 
EQUATIONS SOLVABLE WITH KBSPECT TO X OB y. 15 
 
 substitute for dy, pdx. There will result a differential equation 
 of the first degree between x and p. Solve this equation, and 
 eliminate p between its primitive and the given equation. 
 
 32. Solve the 
 
 following equations : 
 
 1. 
 
 x-yp== ap\ 
 
 2. 
 
 y = xp^ + 2p. 
 
 3. 
 
 (x + ypy = a\l+p') 
 
 4. 
 
 y = xp+p—p'. 
 
 5. 
 
 (y-apy=l+p\ 
 
 6. 
 
 yz=ap-\- bp^. 
 
 7. 
 
 x'-\-y=iy'. 
 
 8. 
 
 f^x^il+p"). 
 
 9. 
 
 y=p' + 1p\ 
 
 33. When the equation can be solved with respect to x. 
 Differentiate, regarding p variable as well as x and y, and 
 
 substitute for dx, -K. There will result a differential equation 
 
 P 
 of the first degree between y and p. Solve this equation, and 
 eliminate p between its primitive and the given equation. 
 
 34. Solve the following equations : 
 
 1 . p^y + 2px = y. 
 
 2. x=p + \ogp. 
 
 3. _p'(a^ + 2ax)= a\ 
 
 4. a^p^ = 1 -hp^. 
 
 5. {x — apy=l+p^; also when a =1. 
 
 6. x = ap + bp'. 
 
 7. my — nxp = yp^. 
 
16 HOMOGENEOUS EQUATIOKS. — CLAIBAUT'S FORM. 
 
 35. When the equation is homogeneous with respect to x 
 owe? y. 
 
 Substitute y = vx. If the resulting equation between p and 
 v can be solved with respect to "y, the given equation comes 
 under Art. 31 or Art. 33. 
 
 But if we can solve with respect to », substitute for », v + x — , 
 
 dx 
 
 and there will result a differential equation of the first degree 
 
 between v and x. 
 
 36. Solve the following equations : 
 
 1. xy'^{p'^ + 2)=2pf + a?. 
 
 2. (2p+l)x^y = x^p'' + 2y^. 
 
 3. 4:x' = 3{3y—px)(y+px). 
 
 4. ds = (^]'dx + ( — ) dw, where ds= Vl +»^- da. 
 
 \2xJ \2yJ 
 
 5. (nx+pyyz=(l+p!')(y^-i.nx^). 
 
 37. Clair aut's Form, 
 
 y=px+fip). 
 
 The solution is immediately obtained by substituting p = c. 
 
 38. Solve the following equations : 
 
 1. y=px + -- 
 
 2. y=px-\-p—p^. 
 
 3. y'^-2pxy-l=p'^{\-a?). 
 
 4. y = 2px+ y'p'. Put y^ = y'. 
 
 5. ayp' + {2x — b)p = y. Put y' = y'. 
 
 6. x'{y—px) = yp^. Put y^=zy', x' = x'. 
 
 7. e'^(p — l)+p'e2>' = 0. Put e' = a;', e'' = 2/'. 
 
 8. (px — y) (py + x) = h^p. Put y^ = y', x^ = x'. 
 
CHAPTER VI. 
 
 SINGULAR SOLUTIONS. 
 
 39. A singular solution of a differential equation is a solution 
 which is not included in the complete primitive. Differential 
 equations of the first degree have no singular solution. Those 
 of higher degrees may have singular solutions, which may be 
 derived either from the complete primitive, or directly from 
 the differential equation. 
 
 40. Let f{x, y, c)= be the complete primitive. 
 
 By differentiating, regarding c as the only variable, obtain 
 
 -=^=0. If we eliminate c between this equation and the prim- 
 dc 
 
 itivBf the result will be a singular solution, provided it satisfies 
 
 the given differential equation. 
 
 41. Let f{x, y, p) =0 be the given differential equation. 
 By differentiating, regarding p as the only variable, obtain 
 
 5ii = 0. If we eliminate p between this equation and the given 
 dp 
 
 differential equation, the result will be a singular solution, 
 
 provided it satisfies the differential equation. 
 
 42. Derive the singular solution of the following equations, 
 directly from the given equation, and also from the complete 
 primitive : 
 
 1. y=px+—- 
 
 2. y'-2xyp + Cl+x^)p^=l. 
 
18 SINGULAR SOLUTIONS. 
 
 3. p= — 4£B2/p + 82/2 = 0. Put J 
 
 4. y = {x — \)p-p-. 
 6. 2/(1 +i>^) = Sk?:). 
 
 6. v?p^-2{xy-2)p + y^ = Q. 
 
 7. {y — xp) (mp — n) = mnjy. 
 
DIFFERENTIAL EQUATIONS OF AN ORDER 
 HIGHER THAN THE FIRST. 
 
 CHAPTER VII. 
 
 LINEAR DIFFEEENTIAIi EQUATIONS. 
 
 General Form, 
 
 d»2/ yd"-!?/ yrf;;^!^ „ dy „ 
 
 TH ^ -Ai , ^ . -r -^-2 • • • -f- -A.„-i-;- + -a.„y — -A, 
 
 the coefficients Xi, Xj, ...X„ and X being functions of x alone 
 or constants. 
 
 43. Linear equations with constant coefficients and second 
 member zero may be solved as follows : 
 Substitute in the given equation, 
 
 — s. = m", i = m"^, -3- = rn, w = m"=l. 
 
 dx" daf'^ dx 
 
 There will result an equation of the nth degree in m, called 
 the auxiliary equation. Find the n roots of this equation ; 
 these roots will determine a series of terms expressing the 
 complete value of y as follows, viz. For each real root mj, 
 there will be a term Ce'"i"' ; for each pair of imaginary roots 
 a±6V— 1, a term e'''{Asmhx + Bco&bx) ; each of the coeffi- 
 cients A, B, C, being an arbitrary constant if the corresponding 
 
 root occur only once, but a polynomial Ci + C2X + CiX^ 1- c^x'"^, 
 
 if the root occur r times. 
 
20 LINEAR EQUATIONS. 
 
 44. Moots of auxiliary equation, real and uneqv^. 
 Solve the following equations : 
 
 2. ^ + I2y=7^. 
 dar dx 
 
 3. a^=^. 
 
 dx' dx 
 
 5. ^+4^ = 2/. 
 dx" dx " 
 
 da? dx 
 
 g_ d?y ^(Py ^ ^dy^ 
 da? da? dx 
 
 9. ^ = 7^-62/. 
 
 10. ^^ + 272, = 12ff. 
 dx* dar 
 
 11. ^_2(a2 + 60^+(a'-&')'— = 0. 
 dar* daf dx 
 
 45. Boots of auxiliary equation unequal, hut not all real. 
 Solve the following equations : 
 
 2. ^_6^ + 132/ = 0. 
 dar dx 
 
LINEAR EQUATIONS. 21 
 
 dsr dx 
 
 (1), when a>b; (2), when a<b. 
 
 dor dx 
 
 6. ^ + 2^ = 0. 
 dx^ dx 
 
 dar dar da; 
 
 da;* " 
 
 10. g + 25-82, = 0. 
 dar dar 
 
 11. ^ + 4a*y = 0. 
 dor 
 
 14. ^ = y. 
 
 dx» ^ 
 
 46. Auxiliary equation containing equal Toots. 
 Solve the following equations : 
 
 ^- ^ ^"d^+"^-^- 
 
22 
 
 LINEAR EQUATIONS. 
 
 2. ^ = 0. 
 
 da? 
 
 3. ^ = 4£y. 
 
 da? da? 
 
 da? da? ^ 
 
 5. 'll-^-^ + y^O. 
 
 da? da? dx 
 
 6. ^ + 2n'^ + n'y = 0. 
 dx^ da? " 
 
 da? da? da? dx 
 
 8. f|_4fl+14g-20^+252/=0, the first 
 
 da;* aar' aar da; 
 
 member of auxiliary equation being a perfect square. 
 
 da;" da;""^ 
 
 47. Linear Equations with constant coefficients and second 
 member not zero. 
 
 There are two methods of solution : 
 
 First. Method of Variable Parameters. — Solve the 
 equation by Art. 43, regarding the second member as zero. 
 
 Supposing it to be of the nth order, this value of y will con- 
 tain n arbitrary constants. Derive from it the successive 
 
 differential coefficients, -^, — f , ^ ; then differentiate the 
 
 da; dar dai^~^ 
 
 values of y, -^, — f , • ^-, regarding the arbitrary constants 
 
 da; dar da;""^ 
 
 alone as variable, and place these n results equal to zero, except 
 
 the last, which put equal to the second member of the given 
 
 equation. These n conditions will determine expressions for the 
 
 n arbitrary constants, which are to be substituted in the original 
 
 expression for y. 
 
LINEAR EQUATIONS. 23 
 
 Second Method. — By successively differentiating the given 
 equation, obtain, either directly or by elimination, a new differ^ 
 ential equation of a higher order with the second member zero. 
 Solve this by Art. 43, and determine the values of the super- 
 fluous constants so as to satisfy the given differential equation. 
 In this last work of determining the superfluous constants all 
 the other constants may be regarded as zero. 
 
 48. Solve the following equations : 
 
 dar dx 
 
 dx* daf daf dx 
 3. ^-a'y=x+l. 
 
 dx' dx' dx 
 
 dx* ^ 
 
 6. '^ + n'y = l+x + a^, 
 daf 
 
 7. — ^- — 2a— + 0,^1/ = €;" ; alsowhena=l. 
 dar dx 
 
 8. — ^ +m^« = cosaa!; also when a = n. 
 dx' '^ 
 
 9. ^_5^ + 62/ = e'«; also when w = 2, or n = 3, 
 dx^ dx 
 
 10. tM-3^+2y = xe''\ 
 dor dx 
 
 11. ^_9^+202/ = a)?e5'. 
 dar dx 
 
 12. — | + 4w = a;sin^ap. 
 dsr 
 
24 LLNEAE EQUATIONS. 
 
 13. --^ + 2—^ + y = x'coaax; also when o=l. 
 
 dx* daf 
 
 14. —-^—'2-f--\-^y = e'cosx. 
 die dx 
 
 49. Linear equations of the form 
 ia+bx)''^+Ma+bxy-^^...+A„_,{a+bx)^+A^=X, 
 
 where Ai, A^, ■■•A„ are constants, and X a function of a; alone. 
 
 Put a +bx = e', and change the independent variable from 
 X to t. The new differential equation between y and t will be 
 linear with constant coeflScients, and may be solved by Art. 47. 
 
 50. Solve the following equations : 
 
 dor dx 
 
 2. (x + ay^-i(x + a)^ + 6y = x. 
 
 dar dx ' 
 
 3. (a + bxy^ + b(a + bx)^ + b'y = 0. 
 
 dor dx 
 
 4. a? — y- — x-^ + 2y = x\ogx. 
 
 dor dx 
 
 5. {2x-\y^ + {2x-l)'^-2y = (i. 
 
 dor dx 
 
 6. l&{x+iyp^ + 96ix + irf^+104{x+iy^ 
 
 dx* dor dor 
 
 + 8{x+l)^ + y = a^ + 4:X + S. 
 dx 
 
 7. a;*^ + eoii'^+9o>^^ + 3x^ + y = il+\ogxy. 
 
 dx* dx^ da? dx 
 
 8. a;2^ - (2m - \)x^ + (m? + n'')y = n'^oTlogx. 
 
 docF dx 
 
CHAPTER VIIL 
 
 SOME SPECIAL FORMS OF DIFFERENTIAL EQUATIONS OF 
 HIGHER ORDERS. 
 
 51. Form, — — = X, where X is a function of x alone. 
 
 ax" 
 
 The expression for y is found by integrating X successively 
 n times with regard to x. Or solve by Art. 47. 
 
 52. Solve the following equations : 
 
 
 
 1. 
 
 x'^y-. 
 
 da? 
 
 = 2. 
 
 
 
 2. 
 
 d'y_ 
 
 1 
 
 
 dx* 
 
 (x + a.y 
 
 
 
 3. 
 
 ^y_ 
 dx" 
 
 of. 
 
 
 
 4. 
 
 d*y_ 
 da^ 
 
 a; cos a;. 
 
 
 
 5. 
 
 da^ 
 
 + 4cosa;=0. 
 
 
 
 6. 
 
 d''y _ 
 da" 
 
 ■■xe". 
 
 
 
 7. 
 
 d?y _ 
 da? 
 
 sin^a;. 
 
 53. 
 
 Form, 
 
 <3Py _ 
 
 T, where F is a fun- 
 
 
 
 da? 
 
 
 
 Multiplying both members by 2-^, and integrating, we have 
 
 (XX 
 
 (^Y= 2 Crdy + Ci. Therefore x = C — --^ 
 
 ■Ci. 
 
26 EQUATIONS NOT CONTAINING y, 
 
 54. Solve the following equations : 
 
 1. 
 
 d^V 2 
 d^ = ''y- 
 
 2. 
 
 dx^ " 
 
 3. 
 
 "da? 
 
 4. 
 
 da? 
 
 5. 
 
 (Vy _ 1 
 
 da? -yjay 
 
 55. Equations not containing y directly. 
 
 By assuming the differential coefflclent of the lowest order in 
 the given equation equal to «, and consequently the other differ- 
 ential coeflBcients equal to the successive differential coefficients 
 of 2 with respect to x, we shall obtain a new differential equation 
 between z and a; of a lower order than the given equation. 
 
 56. Solve the following equations : 
 
 1. a;^ + ^ = o. 
 
 da? dx 
 
 2. ^ = a= + 6Y^Y- 
 da? \dxj 
 
 3. ^-x^=ff'^\ 
 dx da? \da?J 
 
 *■ »'(S)'='+(I' 
 
EQUATIONS NOT CONTAINING x. 
 
 27 
 
 7. 
 
 8. 
 
 9. 
 
 10. 
 
 ^ ' M dx 
 
 dn? dot? \d3? J 
 
 da? da? \ 
 do? \dxj 
 
 d?jl 
 da? 
 
 1 + 
 
 WJ. 
 
 i 
 
 57. Equations not containing x directly. 
 
 By assuming — = a, and consequently 
 dx 
 
 da? dy^ 
 
 do?-'' df^\dy)' 
 
 etc. 
 
 changing the independent variable from a; to 2/, we shall obtain 
 a new differential equation between z and y of a lower order than 
 the given equation. 
 
 58. Solve the following equations : 
 
 1. 
 
 "da? \dxj 
 
 2. 
 3. 
 4. 
 6. 
 6. 
 
 .(l-log.)g+(l+log3.)(|)=0. 
 
 y^ + 
 "dn? 
 
 tMm 
 
 i 
 
 ■dy\\ 
 dx) 
 
 \dx) "da? dx •' \\dx) da?} 
 
 <S)"-(l)=^l-S^"<f)"S- 
 
CHAPTER IX. 
 
 SIMULTANEOUS DIFFERENTIAL EQUATIONS. 
 
 59. Simultaneous differential equations of the first order. 
 There should be n given equations between w + 1 variables. 
 
 Selecting one of these for the independent variable, we may, by 
 differentiating the given equations a sufficient number of times, 
 eliminate all but one of the dependent variables and their diflfer- 
 ential coefficients. The resulting differential equation between 
 two variables must be solved by the methods previously given, 
 and from its primitive and the given equations may be obtained 
 the values of the other dependent variables. The complete 
 solution will consist of n equations containing n arbitrary con- 
 stants. 
 
 In general, if we differentiate the given equations n—\ times 
 successively, we shall have in all m" equations, which are just 
 sufficient for the elimination of w — 1 variables, together with 
 their n(«, — 1) differential coefficients. Shorter processes for 
 the elimination will frequently suggest themselves in special 
 cases. 
 
 60. Solve the following simultaneous equations : 
 
 1. 
 
 d^ : A , y I\ 
 
 — -|-4a;+f- = 0, 
 dt 4 
 
 dt " 
 
 2 -dx= ^y = — ^— . 
 
 3. '^^ = -^y =dt. 
 
 2/ — 7a; 2x + 5y 
 
SIMULTANEOUS EQUATIONS OF THE FIBST ORDEK. 29 
 
 4. 
 
 a h ft V = e 
 
 dec " 
 
 dx 
 
 5. 
 
 ( dx , , , . 
 
 dt 
 
 6. 
 
 dx _ ^y _ 
 
 2y — bx + ^~x — Qy + e'^'^ 
 
 = dt. 
 
 dt dt " ' 
 
 3^ + 7^ + 34a; + 38w = e*. 
 { dt dt " 
 
 '4^ + 9^+lla; + 31w = e', 
 dlt di " 
 
 dt dt -^ 
 
 9. 
 
 ■i^^Q^^2x + 3ly = ^, 
 dt dt 
 
 3^ + 7^ + x + 2iy = 3. 
 dt dt 
 
 10. 
 
 ^ + ].(x + 5y)=t. 
 dt t 
 
 11. 
 
 f tdx= {t — 2x)dt, 
 \tdy={tx + ty + 2x — t) dt. 
 
30 SIMITLTAiTBOUS EQUATIONS OF HIGHER OEDBBS. 
 
 ' dx 
 
 — z=ny — mz, 
 dt " 
 
 12. 
 
 dy 
 dt 
 
 dz 
 dt 
 
 =.lz — nx. 
 
 = mx — ly. 
 
 13. 
 
 It — = mn (y — z), 
 dt ^^ ' 
 
 mt-^=nl(z — x), 
 dt ^ ' 
 
 dz 
 nt— — lm(x — y). 
 dt 
 
 61. Simultaneous differential equations of an order higher 
 than the first. 
 
 By differentiating the given equations a sufficient number of 
 times, we may eliminate all but one of the dependent variables 
 and their differential coefficients, and thus obtain a differential 
 equation between two variables, which must be solved by the 
 appropriate methods. Its primitive, together with the given 
 equation, will enable us to determine the values of the other 
 dependent variables. The general solution will contain a num- 
 ber of arbitrary constants equal to the sum of the highest orders 
 of differential coefficients in the several given equations. 
 
 62. Solve the following simultaneous equations : 
 
 — + n^x = 0, 
 dt' 
 
 — n^x = 0. 
 
 dhj 
 df 
 
 l'^_3a;-42/ + 3 = 0, 
 
 d'y 
 df 
 
 + x + j/ + 5 = 0. 
 
SIMULTANEOUS EQUATIONS OF HIGHER OKDEES. 
 
 3. 
 
 df 
 
 -3a!- 42/ + 3 = 0, 
 
 ^ + a;-82/ + 5 = 0. 
 
 ^x 
 
 + n^3/ = 0, 
 
 
 31 
 
 6. 
 
 2 — f. 4w = 2 a; 
 
 dar da; 
 
 2^ + 4^-3z = 0. 
 ^_ da; da; 
 
 d':r dar da; 
 
 ^ + 4^+(5-7in- + 2(l-nnw = 0. 
 dar* da;2 ^ '^da; ^ "' 
 
 ^_4^ + 4^_2, = 0, 
 da;* da:;' da;^ 
 
 ^_4^ + 4^-2 = 0. 
 dx"* dar dar 
 
CHAPTER X. 
 
 GEOMETRICAL EXAMPLES. 
 
 63. Expressions involved in the examples, p representing 
 
 -^, and q representing — f . 
 dx dar 
 
 y 
 
 Subtangent =-. Subnormal =p2/' 
 
 Normal = y Vl +p^. ~r' = Vl +p'. 
 
 y 
 
 Intercept of tangent on axis of X=x 
 
 Intercept of tangent on axis oi Y=y —px. 
 
 CI -i-rP')^'- 
 Radius of curvature = :f -i — Z_£L£_. 
 
 q 
 
 1. Find the curve whose subtangent varies as (is n times) the 
 abscissa. 
 
 3. Find the curve whose subnormal is constant and equal 
 to 2 a. 
 
 3. Find the curve whose normal is equal to the square of the 
 
 ordinate. 
 
 4. Find the curve for which s = ma?. 
 
 5. Find the curve for which s? = y^ — a^. 
 
 The orthogonal trajectory of a series of curves is a curve 
 that intersects them all at right angles. 
 
 Describe the curves represented by the following equations, 
 and find their orthogonal trajectories : 
 
 Q. y z= mx, m being the variable parameter. 
 
GEOMETRICAL EXAMPLES. 33 
 
 7. y" = 2 OX — a^, a being the variable parameter. 
 
 8. ^^ = 400;, a being the variable parameter. 
 
 9. xy = k', k being the variable parameter. 
 
 10. — 4- 2^ = 1, b being the variable parameter. 
 
 (X 
 
 11. x' + m'y' = m^a^, a being the variable parameter. 
 
 12. — — - + ^ = 1, b being the variable parameter. 
 b' + /r b' 
 
 The three following examples require the singular solution : 
 
 13. Find the curve such that the sum of the intercepts of the 
 
 tangent on the axes of X and yis constant and equal to a. 
 
 14. Find the curve such that the part of the tangent between 
 
 the axes of X and Fis constant and equal to a. 
 
 15. Find the curve such that the area of the right triangle 
 
 formed by the tangent with the axes of X and Y is 
 constant and equal to a". 
 
 The following examples require the solution of differential 
 equations of the second order : 
 
 16. Find the curve such that the length of the arc measured 
 
 from some fixed point of it is equal to the intercept of 
 the tangent on the axis of X. 
 
 17. Find the curve whose radius of curvature varies as (is n 
 
 times) the cube of the normal. 
 
 18. Find the curve whose radius of curvature is equal to the 
 
 normal ; first, when the two have the same direction ; 
 second, when they have opposite directions. 
 
 19. Find the curve whose radius of curvature is equal to twice 
 
 the normal ; first, when the two have the same direction ; 
 second, when they have opposite directions. 
 
AI^SWERS. 
 
 Art. 3. (p==^ 
 \ ax 
 
 1. y{\+x)+px{l — y) = Q. 6. y=px+p—p\ 
 
 2. {x' + l)pxy = y^ + l. 7. 053- = a. 
 
 3. ta.ux=ptany. 8. p^ + 2pycotx = y'. 
 
 4. (l+a;')p+3/ = tan-'a;. 9. x'p^ = l+p\ 
 6. (2/loga;— l)y =px. 
 
 Art. 5. 
 
 1. fS + a^y = 0. 4. x^||_xf^=3y. 
 dar dor dx 
 
 2. — | — a^w = 0. 5. — ^ + n^y = cosax. 
 dor da? 
 
 3. ^_2a^ + a^y = 0. 
 dar da; 
 
 Art. 7. 
 
 1_ ^ = 'j^_ey. 4. ^_3^+3^_^=0. 
 
 da;' da; da;* da;' da;^ dx 
 
 2. ll + ^ + ^^^y. 5. ^_a*« = a;'. 
 da;' dar' da; da;* 
 
 3. ty-2^ + ^ = e'. 
 da? da? dx 
 
 Art. 9. 
 
 1. log (an/) 4- a;- 2/ = c. 3. (1 + ar^) (1 + j/^) = car*. 
 
 2X+V V A 1 - 
 
 • — — ^+log- = c. 4- x'y = ce''. 
 
 xy ^ ^x 
 
36 
 
 ANSWERS. 
 
 5. log[(2/ + Vr+?)Vl+2/''] = 
 
 . + c. 
 
 6. coBj^ = ccos,a;. 
 
 7. tauxtan^ = c. 
 
 1. y = ce ". 
 
 2. 2/ = ce-V'|. 
 
 3. y — ce'. 
 
 4. a^ = c^ + 2«/. 
 
 8. sin'' «+ sin^y = c. 
 
 9. a;^— l = c(a; + 2/ + l). 
 
 Art. 11. 
 
 5„ — fiin^ 
 
 . a; = ce * . 
 
 6. {y + xy{y + 2xy = c. 
 
 7. log (a^ + 2/0 = 2 tan-i^'+ c. 
 
 8. a;2/cos^=c. '^ 
 
 9. (1), log(a^-ma;i/ + 2/0 + 
 
 2m 
 
 V4 — m* 
 
 rtan 
 
 -I 2y — ma; _^ 
 a V4 — m^ 
 
 (2), x — y = ce"". 
 
 (2 y — ma: + a: Vm" — 4) 
 
 m— "SJ mr —4 
 
 (3), 
 
 (2 y - ma; - a;Vm' - 4)"'+^"''-' 
 
 10. 2/ sin a — a; cos a + Va;^ + 2/^ = c (ar' + y') . 
 
 Art. 13. 
 
 1. (2/_a; + l)2(2/ + a:-l)' = c. 
 
 2. a; + 2y + log(2a; + ?/— l)=c. 
 
 3. a; + 52/ + 2 = c(a;-y + 2)*. 
 
 4. 4a; — 8y = log(4a; + 8?/ + 5) +c. 
 
 6. log[2(3a;-l)'+(3y-5)n-V2tan-' ^^^^^~^^ =c. 
 
 ay — 5 
 
1. y = caf-\- 
 
 1 — a a 
 
 ANSWERS. 
 Art. 15. 
 3, y = 
 
 37 
 
 ■ce 
 
 n/w 
 
 2. 2/ = «« + ca; Vl — a^. 
 
 4. 2/ = 8ma; — 1 +ce" 
 
 5. a; = tan^^j/ — 1 + ce" 
 
 tan ^y 
 
 6. (a;+ Va2 + a^)y=anog(K+ Va^ + ar') + c. 
 
 7. yVl+a^ = log ^^+'^~ V ( 
 
 X 
 
 fl. 2/ = ce"* + <^ — 1. 
 
 Art. 17. 
 
 1. ?/ = (cVl-a^-a)-'. 3. y = 
 
 ce2'^ + «(2«2 + l)1 *. 
 
 2. v' = ce* 
 
 a; + l 
 
 SI? 
 .2 
 
 5. y-"+^ = ce'"-"'"""^ + 2 sin a; -+ 
 
 2 
 
 m-1 
 
 6. y = (cx + loga; + 1)~'. 
 
 7. a,= (!L±2)yf 
 
 8. y = 
 
 tana; + sec a; 
 
 sina; + c 
 9. (462 + l)y2 = 2a(sina; + 26eosa;) + ce-"^ 
 
 Art. 19. 
 
 1. x'' + Qa?f + y^ = c. 6. a? + y'' + 2ta.xr^l,= c. 
 
 2. a;^-6»22/-6a;/ + 2/2 = c. 6. y'' = c--2cx. 
 
 3. a;'-2/2=ca;. 7. »= + y^ + 2 sin-^| = c. 
 
 Z 
 
88 ANSWERS. 
 
 9. e'(x' + f) = c. 
 
 10. cos {mx + ny) + sin (nx + my) = c. 
 
 11. VTT^+P + tan-i- = c. 
 
 12. (1), log?l^^±i^=2W^^, 
 
 x'Vg — yVb « 
 
 (2), tan-'^:^+^!:^ = c. 
 
 (4), ^_-^y^_ = ca^^K 
 
 (5), ^ = c. 
 
 a g^x* 
 
 Art. 22. 
 
 1. 3^ + 3/^ = 0(3; + 2/). 4. «2 + 22/2 = cVar' + /. 
 
 2. a;^ — y' + xy = c. 6. 2/ = ex. 
 3.3/ = ca;. 
 
 (3), c«-A-^l=c. 
 
 Art. 24. 
 
 I 
 
 , - . 2 , CO! 
 
 1. a; = eye*. 4. -7= = log— ' 
 
 Va;.v 2/ 
 
 2. y = ce". 5. aey = c. 
 
 3- an/ - — = logC2/2. 6. a;2/e^ = log^. 
 
 7 . a;?/ + log sin (a;?/) = log — 
 
 Art. 26. 
 
 1. e'(ar' + 2/2)=c. 3. x'-f^cx. 
 
 2. a?-f = Ctf. 4. 2/ Vl- 3)2 + 35(1 -3/) = 6. 
 
ANSWEE8. 39 
 
 6. sin a cos 3/ + 2/ tan 2a; = c. 6. sin^a;sin2 (a — y) = c. 
 7. y = e-J''^(jQj''^dx + c\ 
 
 Art. 28. 
 
 X 
 
 1. {x — a){y — b){z — c) = c'. 7. « = ce». 
 
 2. ar'+2?/^-6a,-2/-2a;a+2^=c. 8. a;2/ = c(a + «). 
 
 2 
 
 3. yz + zx + xyz=c. 9. a; = \-c. 
 
 y + ci 
 
 4. xyz = c. 10. 2/(a; + «; = c(!/ + 2). 
 
 5. e'(?/ + «) = c. 11. e^(a; + y + 22) = c. 
 
 6. « + ^ + «=c'. 12. 2^%l+^ = c. 
 X y z X y 
 
 Art. 30. 
 
 1. (2/-2a; + c)(2/-3a! + c) = 0, or {bx-2y + cy = a?, 
 
 2. (2^ + c)» = a2(loga;)^ 3. (3/+c)2 = 4ax. 
 
 *• (2/ + c)* = (Vx - ar" + sin"' Va;)''. 
 
 5. {xy + c)(3?y + c)=Q. 
 
 6. (a;'-2i^+c)[e'(a; + y-l) + c]=0. 
 
 7. {y + c){y + x' + c){xy + cy+l) = Q. 
 
 8. (a3-3i/ + c)(e2+«/)(an/ + C2/ + l)=0. 
 
 9. y'^sa?x-\-1cy-\-<? = Q. 
 
 Art. 32. 
 
 1. Eliminate p by means of x = — =r (c + asin^'p) , 
 
 Vl— p^ 
 
 2. Eliminate J3 by means of a;(^ — 1)' = logp'' — 2p + c. 
 
 3. Eliminate » by means of a; = , (c H 1- atan~'» )• 
 
40 
 
 ANSWERS. 
 
 4. y = cx-i- c — (^. 
 
 5. X = alog (ay ± Va^ + y'—l) + log (y :f y/a^ -\-y^ — \) + c. 
 
 6. a; ± -\/ciF+Tby = alog (a ± V«^ + 4 6y) + c. 
 
 ± Va;-' + 2/ + a5(vT7 + l) 
 y + Vy^ — afY 
 
 8. 
 
 2/ Vy^ — 0? — a:^log 
 
 = (2/2 + a^logcw2)2 
 
 9. 4 (a; + c)=+ (a; + c)^ - 18y (a; + c) ^ 272/'' - 42/ = 0- 
 
 Art. 34. 
 1. y^='icx-\-(?. 
 
 2. a;+l=± V2^ + c4-log(± V22/ + C-1). 
 4. e^>'4-2ca;e'' + c^ = 0. 
 
 a^— 1 
 when a=l, 4y = a^ — log (ca^). 
 
 6. 62(6y + c)2+(6a&a;+a')(62/ + c)-3aV-16 6a^ = 0. 
 
 7. c(nar*+2y2±a;V7iV+4m/)"= [(2m-m)a!± VnV+4my2]'> 
 
 Art. 36. 
 
 1. (a;2 _ 2/2 + c) (ar^ - 2/' + ca!*) = 0. 
 
 2. (V«— Vy + c)(VK — V^ + ca;) = 0. 
 
 3. a;^ + 2ca;V32/2-a^-c2=0. 
 
 4. (2/_a;)^±*'2^c(V^ + Va)'- 
 
 U 1 
 
 5. cc2'— 2ca^~W2/^ + ria^ + wc2 = 0, where A; = -ii 
 
ANSWERS. 
 
 41 
 
 1, m 
 . y=cx-\ 
 
 2. y = cx + c — (^. 
 
 3. (y-ca;)''=l+c='. 
 
 4. y' = ex + — • 
 " 8 
 
 Art. 38. 
 
 5. 42/^= 2 c(2a;- 6) + ac^. 
 
 6. 2/^=ca^ + c^ 
 
 7. e''=ce' + c\ 
 
 8. 2/2-ca^ = - 
 
 c/i^ 
 
 c + 1 
 
 Art. 42. 
 
 Complete Primitives and Singular Solutions : 
 
 1 I '"' 
 
 1. 2/ = ca; + _, 
 
 c 
 
 2. (y-cxy=l-c', 
 
 3. 2/ = c(a! — c)^ 
 
 4. 2/ = c(a; — 1) — c^, 
 6. 2/^-2cx + c2=0, 
 6. (j/ — ca;)^=^4c, 
 
 2/^ = 4 mx. 
 
 7. (y — ca;) {mc — n)= mnc, I—] 
 
 y"^ — a? =1. 
 
 ^ 27' 
 42/=(a;-l)2 
 
 xy= 1. 
 
 = 1. 
 
 1. y = CiB"" + c^e-'^ 
 
 2. y = c^e^ + c^e*"- 
 
 X 
 
 3. y = Cie' + C2. 
 
 X 
 
 4. y = Ci^'' + CiCS . 
 
 Art. 44. 
 
 5. ye^' = Cje'^' + Cje-'*^^ 
 
 oz bx 
 
 6. y = Cie' ^M". 
 
 7. y=-c,e''' + c,e-'^+Cs. 
 
 10. 1/ = Cje'' + c^e-^' + Cae^*'" + C4e-''^3- 
 
 11. y = de*""""' + Cse"-"'"^ + 036'''+"^ + 046-'"+"' + Cg. 
 
42 
 
 ANSWERS. 
 
 1. 
 
 2. 
 3. 
 
 4. 
 5. 
 6. 
 
 7. 
 
 8. 
 
 9. 
 10. 
 11. 
 12. 
 
 13. 
 
 Art. 45. 
 
 y = CiSma; + C2Cosa;. 
 
 2/ = e'"" (ci sin 2 a; + Cj cos 2 x) . 
 
 "When a > 6, ?/ = e'"=^(cie^ V"^^' + Coe-^V^^^^) ; 
 
 when a<b, y = e'"(ci sina; V6^ — a^ + c^cosx y/V — a?) . 
 y = ^"^ [ci sin {a?-W)x + c,cos {a' - b')x] . 
 y = a"(ciSina;4-C2Cosa;). 
 ?/ = CisinxV2 +C2eosa; V2 + c^. 
 
 y^ae' + e 2fc2sm-— — + C3COS— — j- 
 
 ye'' = Cy e'"' + Cj sin x V2 + C3 cos a; V^. 
 
 2/ = Cie' + c^er' -\- C3sina; + Cicosa;. 
 
 2/ = de""^ + 026"==^^ + C3sin2a; + C4COS 2a;. 
 
 y = e" (ci sin aa; + Cj cos ax) + e"" (Cj sin aa; + C4 cos ax) . 
 
 y = Cifi^^ + Cje-' + (036= + €46 2) sin 
 
 V3 
 
 + (Cje^ + cse 2) cos 
 
 2 
 a;V3 
 
 2 ( • a; , a; \ 
 
 2/ = CiSma! + C2COsa; + e ( C3Sin- + C4Cos- 
 
 + e 
 
 »v;3 
 2 
 
 C5sin| + CeC0s| 
 
 14. 2/ = Cje^+Cje^'+Cgsinaj + CiCosaj+e^'YcsSin^ f-cecos — 
 
 V2 
 
 V2, 
 
 - ■ / 
 
 + e *'2/(^siQ — -^Cjcos ). 
 
 V V2 V2y 
 
 1. y= (ci + C2a;) e"". 
 3. 2/ = Cie*' + C2 + C3a;. 
 
 Art. 46. 
 
 2. 2/ = Ci + C2a;. 
 
 4. y = Cye-''+{c2 + c^x)e- 
 
AKSWERS. 43. 
 
 5. y = Cie-'+ (c^ + Csx)^. 
 
 6. 2/ = (Ci + C2X) cosnx +(03 + c^o;) sinnx. 
 t- y={ci + eiX + Csa?)e + Ci. 
 
 8. 2/ = e'[(Ci + C2a;)sm2a;+ (03 + 040;) cos 2 a;]. 
 
 9- 2/ = Ci + CiX + c^a? ■•• + c„_2a;"-' + c„_isina; + c„cosa!. 
 
 Art. 48. 
 
 ^ 14i 
 
 2. 2/ = Cisina; + C2Cosa; + (c3 + C4a;)e* + a. 
 
 3. 2/ = Cie'° + C2e-"-5±i. 
 
 a'' 
 
 4- y^Ui + CiX+^^e' + Cs. 
 
 5. 3/ = Cie°' + C2e~'" + C3smaa; + C4COsaa; 
 
 a* 
 
 B ■ , , 1+x + x^ 2 
 
 6. w = Ci sm wx + Cj cos na; + • — - — r-! 
 
 7. w = (ci + C2a;)e'"H ^^; 
 
 when a—1, y=(ci + C2X + —\e'. 
 
 o . , , COS oa; 
 
 8. w = CiSmwa; + C2C0sria;+— -; 
 
 n' — ar 
 
 , . , , xsinnx 
 
 when a = n, y = Cj sin na; + C2 cos »ia; + 
 
 9. y = Cie2' + C2e^"' + 
 
 2n 
 
 w'i-Sn + e 
 when n=2, y=:(ci — x)e'' + C2^''; 
 when n=3, y = Cie^'' + (c2 + a;)e'*- 
 
 H '^ -r !> ^„2_3,,^2 (»i=-3n + 2)" 
 
44 ANSWBKS. 
 
 11. y = c^e*' + C2e='+?■^-±^^-±-^e'^ 
 
 * ^ ley V 32/ 8 
 
 13. y = (ci + C2a;)sina!4-(c3 + C4a;)cosa; 
 
 , r a? 4(5a2+l)"l 8aa; 
 + i ^ ■' cosaa; -sinaa;; 
 
 when a=l, ?/= ^1 + 020; + — — ^jsina; 
 
 + /"ca + C4X - — + — + -Vosa;. 
 
 14. w = Ci e-2' + f C2 + -^ - — \"= cos a; + f C3 - — + ^V' 
 
 " \ 4.00 20) \ m 20) 
 
 Art. 50. 
 
 1. y = c,^ + ^- 
 
 D 
 
 3. y = Cisin log (a + 6a;) + Czcos log (a + fta;) . 
 
 sin 07. 
 
 4. ?/ = a;(cisin log a; + C2COS log a; + log a;) . 
 
 5. y = {2x-l)\_c^ + c,{2x-\y +c,{'2x-l) ^J. 
 
 6. y = [c. + C2log(a;+l)]V^+'l+ «3 + cJog(^+l) .. ^ 
 
 VaT+l 
 7- 2^ = (ci + Czlog a;) sin log a; + (C3 + Ciloga;) cos log a; + (log a;)' 
 
 + 21oga; — 3. 
 8. 2/ = a;'"(C] sin loga;" + C2 cos log a;" + logo;) . 
 
 Art. 52. 
 
 1. 2/ = Ci + C2X + C3»^-t-a^loga;. 
 
 2. y = Ci + Ci!>i + c^a? + Cia? — {x + ay\og^x + a. 
 
ANSWERS. 45 
 
 3. y=Ci + C2X + C3X^ l-C„!B"-'+, , 
 
 |m-f- w 
 
 4. y = Ci + C2X + CgX^ + Cia^ + a;cosa; — 4 sino;. 
 
 5. y = Ci + C2X + €3X^ + 0^0? + e~' cos X. 
 
 6. 2/ = Ci + c^x + c^x^ \- c„x'*~^ + (x — n)e'- 
 
 m , , 2 , 7cosa! cos'a; 
 
 I 27 
 
 Art. 54. 
 
 1. aa;=log(2/+v'2/^ + Ci) + C2, or y = c'le'" + c'^e 
 
 2. ax = sin"'?- + Ca, or y = Ci sin {ax + Cj) ■ 
 
 3. (cix + c^y^a + ciy^. 
 
 4. a;V2n = Cilog — ^ ■ hca. 
 
 5. 3a;=2a*(y*-2ci)(y^ + Ci)^4-C2. 
 
 Art. 56. 
 
 1. y=Cjloga; + Cj. 
 
 2. 6''y = logsec[a&(x + Ci)] +C2. 
 
 3. 2, = ^ + x/(cO+C2. 
 
 2?/ - 1 -- 
 
 4 zl = (^e' +.Ci e » + C2. 
 
 a 
 fi. (x + c,y+{y + <hy = a\ 
 
 6. 2?=Cia;+(Ci»+l)log(a:-Ci)+C2. 
 
 7. y = Cisin-'a;+ (sin~*a!)^ + C2. 
 
 8. y = -^{x + c,^a^y + CiX + C3. 
 
 15 Ci 
 
46 ANSWBES. 
 
 9. 12y = vf + CiW — Gw\ogw + c2, where w = a; + C3. 
 
 10."!^ Vc^ = w ■\/w^ + c/ + Ci^Og (w + Vw^ + Ci^) + C2, 
 where w = a; + C3. 
 
 Art. 58. 
 
 1. f = x' + CiX + C2. 4. Cia; = log[ci?/+/(Ci)]+Cj. 
 
 2. logy— 1 = 5. logy = Cie' + C2e-". 
 
 CiX -\- C2 
 
 3. Ci2/ = C2eV— Vl+a^Ci". 6. y"+i = Cie''=' + ( 
 
 
 Art. 60, 
 
 1. 
 
 
 3. {^^ = 
 {.y = 
 
 j?/ = — 2cie-' + C2e-'', 
 
 2a; = e-'"[(Ci + Cj) sinJ + (C2 — Cj) cosi], 
 2/ = e" ^' (Ci sin i + C2 cos i) . 
 
 n' — 1 
 
 *• ■ e^ 
 
 a2 = — ncie"' + wc2e""' r — , 
 
 n^ — 1 
 
 -. = (c, + C20e--|^4--, 
 2/ = - (ci + C2 + Cjf ) e-" + -gg- + 25* 
 •a;=2c,e--C2e- + |^+g, 
 
ANSWERS. 
 
 47 
 
 _, , _6, 29e' , 19t 56 
 
 9' 
 
 - '7 3 9 
 
 / , ,N -4< 496^' , 31e' 
 ^ ' ' ^ 36 25 ' 
 
 y = - (ci + c, + cOe-*' + 1||1' - il|l. 
 
 00 z5 
 
 10. 
 
 r a; = (cisini + c^cost) «"*' + — - — , 
 ^ ^ 26 17' 
 
 2e' 
 
 2/= [(C2-Ci)sin«- (c2 + Ci)cos«]e-*'-— - + — . 
 
 lo 17 
 
 ' a; = Ci r" + 2 C2 r' + — + — , 
 10 15 
 
 « = - Ci«-< - c^r^ _ A + ll'. 
 
 - 20 15 
 
 11. 
 
 12. 
 
 ' X = Cif -2+ -, 
 
 y=C2e'-c,r''--- 
 
 X = OiSvakt + a2 cos fcf + osj, 
 y = bi sin &i + 62 cos A;< + 63, 
 z = Ci sin &< + C2 cos kt + c^, 
 where fc^ = ? + m^ + w^. 
 
 The arbitrary constants are connected by the following equa- 
 tions : 
 
 mci — nbi nai — Ici Ibi — ttMi 
 
 C2 
 
 '■ = Tc, 
 
 I m n 
 
48 ANSWERS. 
 
 fx = aisin (klogt) +05008 (ifclog<) + Og, 
 y = bi sin {k log t) + dj cos {k log t) +bg, 
 z = Ci sin (A;log<) + C2 cos (Aslogf) + Cj, 
 where ^^ = 1' + m' + 'n?. 
 
 The arbitrary constants are connected by the following equa- 
 tions : 
 
 mw (Ci — &i) _ wZ(a]— Ci) _ Im (&i — ^i) _ r. 
 
 Pai + m''6i + w^Ci = 0, 03 =63 =03. 
 
 Art. 62. 
 
 0; = Cisinnt + CjCOsnt, 
 
 
 a; = (Ci + C2O e' + (C3 + c^i) g-' - 23, 
 
 g I a; = (Ci + Cjf) e- + 
 1 — 2?/= (ci — C2 + 
 
 C20e'+ (C3 + Ci + C4f)e-' — 36. 
 
 ' X = 4cie" + 4026-=" + Cac'^' + 046-'*" + i, 
 
 3- ^ 9 ' 
 
 14 
 
 a;=e-f cisin4+ "^cos ^^Vg-^f c,sin ^+C4Cos ^\ 
 \ V2 V2y V V2 V2y 
 
 — c^sin^^+c.ooa 
 
 V2y 
 
 T7-J . nx. nx\ -^/ . nx , 91a; \ 
 
 y=:e*'^( Cjsin --— CjCos -^ +« *'^( — Cism —=+ C3COS — \. 
 
 I \ V2 V2/ V V2 -> 
 
 8x 
 
 a; 
 
 2/= (ci + C2a;)e' + 3c3e ^ - g' 
 
 5. < 3^ 1 
 -2= 2(3c2 — Ci— Cja!)^' — 036 ' ~o' 
 
 6. 2/ = w + 'y, « = — « + ■«, 
 
 where u = Ci e^" + (cs e"' + C3 e"'") ^, 
 v = Cie~'' + (056""^+ Cee""')e~' 
 
ANSWERS. 49 
 
 7. y=:u + v, z = u — v, 
 
 where m = (cj + Cj a; + C3 e^^* + C4 6"'*'=') e", 
 
 Art. 63. 
 
 1. x = cy'\ 
 
 2. y^ = iax + c, a parabola. 
 
 3. ±(a; + c)=log(2/ + V2/^-l), 
 
 or y = :^(^+'-\- €'"-'), a catenary. 
 
 4. imy + c = 2'mx'\/4'm,^a? — 1 + log (2 ma;— ■VimJ'x' — 1). 
 
 6. ±(a; + c) =alog(y + -vY^^), 
 
 ±(a; + c)=alog^±^^iEZ, 
 a 
 
 from which y = ^ie'+e "J, a catenary. 
 
 6. a^ + y' = c', a circle. 
 
 7. ar^ + ^ — 2cy = 0, a circle. 
 
 8. 2ar' + /=2c2, an ellipse. 
 
 9. x' — y' = c^, an equilateral hyperbola. 
 
 10. j/* + «^=aMogar' + c. 
 
 11. y = cx'^^. 
 
 12. -^ =1, an ellipse or hyperbola. 
 
 /r — c- r 
 
 13. x^ -\-y^ = a^, a parabola. 
 
 14. x^ +y^ = a*, a hypocycloid. 
 
 15. 2a;y=a^ an equilateral hyperbola, 
 
 16. c^^''— log2/''=4c(a! + c'). 
 
60 
 
 ANSWERS. 
 
 „2 
 
 17. cy" _ — (a; + c')^ = 1 , a hyperbola, when n > ; an ellipse 
 
 n 
 
 or hyperbola, when n < 0. 
 
 18. First, (a; + c')^ + 2/^ = c2, a circle. 
 
 Second, ± (a; + c') = clog (y +Vy^ — c^) , 
 or ± (a; + c') = clog ^ + ^^' ~ — , 
 
 c 
 
 from which y = -le ° + e ° j, a catenary. 
 
 19. First, w + c' = cvers"'' V2 cy — y^, a cycloid. 
 
 Second, (x + c')' = 2 q^ — (?, a parabola. 
 
EEEATA. 
 
 p. 9, Ex. 12, for =x''-''dx read = — x''-'dx. 
 
 p. 11, Ex. 5, for 2yseox read 2ysecy. 
 
 p. 21, Ex. 5, for l+(loga)2 read [1 +(loga)2]y. 
 
 p. 36, Art. 11, Ex. 7, for 2tan-'^ read 2tan-'-. 
 
 X y 
 
 p. 40, Art. 32, Ex. 7, for ± V^+J - ^-(VT?- 1) 
 
 ±Vo?+y+x{Vl7 + l) 
 read ± 4 V^4^-a;(Vl7 - 1) _ 
 ±4Va^+2/ + a;(V'l7 + l)" 
 
 p. 40, Art. 34, Ex. 3, for (w + c) == = ^^-±-^' 
 
 X'+2ax 
 
 read e« + 2ce;(a; + a) + aV = 0. 
 p. 41, Art. 42, Ex. 6, for 4c read —4c. 
 
 p. 42, Art. 45, Ex. 14, for ^ and - — 
 
 V2 V2 
 
 read — r and =:• 
 
 V'2 V2 
 
 p. 44, Art. 48, Ex. 13, erase -- and +-• 
 » ' 4 8 
 
 p. 44, Art. 48, Ex. 14, erase +— and -— • 
 
 400 80 
 
 p. 44, Art. 50, Ex. 6, for <^s+ c.log (x + 1) 
 
 Vx + i 
 
 read ^3 + C4log(a;+ 1) , x'+bJx + ^l 
 
 Vx + 1 225 
 
 p. 45, Art. 54, Ex. 3, for a read — a. 
 
 p. 46, Art. 56, Ex. 10, for yVc^ read 22/Vci. 
 
 p. 48, Art. 62, Ex. 4, in second members, for x read t. 
 
MATHEMATICS. 75 
 
 A Treatise on Plane Surveying. 
 
 By Daniel Caehart, C.E., Professor of Civil Engineering in the 'West- 
 ern University of Pennsylvania, Allegheny. Illustrated. 8vo. Half 
 leather, xvil + 498 pages. Mailing Price, $2.00; for introduction, $1.80. 
 
 rpHlS work covers the whole ground of Plane Surveying. It 
 illustrates and describes the instruments employed, their ad- 
 justments and uses; it exemplifies the best methods of solving the 
 ordinary problems occurring in practice, and furnishes solutions 
 for many special cases which not infrequently present themselves. 
 It is the result of twenty years' experience in the field and technical 
 schools, and the aim has been to make it extremely practical, having 
 in mind always that to become a reliable surveyor the student needs 
 frequently to manipulate the various surveying instruments in the 
 field, to solve many examples in the class-room, and to exercise 
 good judgment in all these operations. Not only, therefore, are 
 the different methods of surveying treated, and directions for using 
 the instruments given, but these are supplemented by various field 
 exercises to be performed, by numerous examples to be wrought, 
 end by many queries to be answered. 
 
 Chapter I. Chain Surveying. 
 
 II. CompasB and Transit Surveying. 
 
 III. Declination of the Needle. 
 
 IV. Lajring Out and Dividing Land. 
 V. Plane Table Surveying. 
 
 VI. Gtovemment Surveying. 
 
 VII. City Surveying. Including the Principles of Levelling. 
 VIII. Mine Surveying. Includingthe Elements of Topography. 
 
 The following Tables have been added: Logarithms of num- 
 bers ; Approximate equation of time ; Logarithms of trigonometric 
 functions; For determining with greater accuracy than the pre- 
 ceding ; Lengths of degrees of latitude and longitude ; Miscellaneous 
 formulas, and equivalents of metres, chains, and feet; Traverse; 
 Natural sines and cosines; Natural tangents and cotangents. 
 
 The judicial functions of surveyors, as given by Chief Justice 
 Cooley, are set forth in an appendix. 
 
 As a practical and complete treatise, Carhart's Surveying has 
 received a cordial welcome. 
 
MATHEMATICS. 
 
 81 
 
 A Short Table of Integrals. 
 
 Beyised and Enlarged Edition. To accompany Byerly't Integral Cal- 
 culus. By iJ. O. Pkikce, Jr., Instructoi in Mathematics, Harvard Uni- 
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 Byerly's Syllabi. 
 
 By W. E. Bteblt, Professor of Mathematics in Harvard University. 
 Each, 8 or 12 pages, 10 cents. The series includes, — 
 
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 Elements of the Differential and Integral Calculus. 
 
 With Examples and Applications. By J. M. Taylor, Professor of 
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 The chapter on diSerentiation is followed by one on direct integra- 
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82 
 
 MATHEMATICS. 
 
 Elementary Oo-ordinate Geometry. 
 
 By W. B. Smith, Professor of Physics, Missouri State University. 12mo, 
 Cloth. 312 pages. Mailing Price, $2.15; for introduction, $2.00. 
 
 TKTHILE in the study of Analytic Geometry either gain of 
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 Elements of the Theory of the Newtonian Poten- 
 
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MATHEMATICS. 
 
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 Academic Trigonometry : piane and sphencai. 
 
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 University of Des Moines. 12mo. Paper. 33 pages. Mailing Price, 
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 fyiHE Plane and Spherical portions are aiTanged on opposite pages. 
 
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 setts Institute of Technologyi Boston. 12mo. Cloth, vii + 50 pages. 
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84 
 
 MATHEMATICS. 
 
 Analytic Geometry. 
 
 By A. S. Hasdt, PhJ!)., Professor of Mathematics in Dartmouth College, 
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MATHEMATICS. 85 
 
 Elements of Plane Analytic Geometry. 
 
 By John D. Bvnele, Walker Professor of Mathematics in the Massa- 
 chusetts Institute of Technology, Boston. 8to. Cloth, ii + 311 pages. 
 Mailing Price, ©2.25 ; for introduction, $2.00. 
 
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 might be met with by the draughtsman at any time, showing the 
 application of the principles of Descriptive Greometry, a feature 
 hitherto omitted in text-books on this subject. AU of the prob- 
 lems have been treated clearly and concisely. The author's sole 
 aim has been to present a work of practical value, not only as a 
 text-book for schools and colleges, but also for every draughtsman. 
 The contents are: Chap. I., Elementary Principles ; Notation. 
 Chap. II., Problems relating to the Point, Line, and Plane. Chap. 
 III., Principles and Problems relating to the Cylinder, Cone, and 
 Double Curved Surfaces of Revolution. Chap. IV., Intersection of 
 Planes and Solids, and the Development of Solids; Cylinders; 
 Cones ; Double Curved Surfaces of Revolution ; Solids bounded by 
 Plane Surfaces. Chap. V., Intersection of Solids. Chap. VI. Mis- 
 cellaneous Problems. 
 
iXMNaiUNIVERSrTYLBRARY 
 
 OCT 20 1993 
 MATHBMTCSUBRABY 
 
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