%n^" 
 
 Out.-^ 
 
EXAMPLES 
 
 DIFFERENTIAL EQUATIONS 
 
 RULES FOR THEIR SOLUTION. 
 
 BY 
 
 GEORGE A. OSBORNE, S.B. 
 
 Professor op Mathematics in the Massachusetts Institute 
 OF Technology. 
 
 o>&<o 
 
 BOSTON, U.S.A.: 
 
 PUBLISHED BY GINN & COMPANY. 
 
 189 9. 
 
Entered, according to Act of Congress, in the year 1886, by 
 
 GEORGE A. OSBORNE, 
 in the Office of the Librariaci of Congress, at Washington. 
 
 J. S. CusHiNG & Co., Printers, Bostow. 
 
06 
 
 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 all 
 been tested by use in the class-room. 
 
 G. A. Osborne. 
 
 Boston, Feb. 1, 1886. 
 
 n^30f>150 
 
Digitized by the Internet Archive 
 
 in 2008 with funding from 
 
 Microsoft Corporation 
 
 http://www.archive.org/details/examplesofdifferOOosborich 
 
CONTEI^TS. 
 
 CHAPTER I. 
 
 DEFINITIONS. DERIVATION OF THE DIFFERENTIAL EQUATION FROM THE 
 
 COMPLETE PRIMITIVE. 
 
 PAGE. 
 
 Definitions 1 
 
 Derivation of differential 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'Y^dy^O '. 4 
 
 Homogeneous equations 5 
 
 Form, (ax -\- hy ■\- c) dx ■\- (a^x ■\-h'y ■\- d^dy = 5 
 
 Linear form, -^ ■\- Py = Q 6 
 
 dx 
 
 Form, ^+Py=Qy- 7 
 
 dx 
 
 CHAPTER III. 
 
 EXACT DIFFERENTIAL EQUATIONS. — INTEGRATING FACTORS. 
 
 Solution of exact differential equations 8 
 
VI CONTENTS. 
 
 PAGE. 
 
 Solution by means of an integrating factor in the following cases : 
 
 Homogeneous equations 9 
 
 Form, f^{xy)ydx-\-f^{xy)xdy = (> 10 
 
 dM_dN dN dM 
 
 When iL_^ = K-), or±^=Hy) 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 a: 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. 
 
 SPECIAL FORMS OF DIFFERENTIAL EQUATIONS OF HIGHER ORDERS, 
 
 PAGE. 
 
 Form, ^=X 25 
 
 Form, 4^ = r 25 
 
 Equations not containing y directly , 26 
 
 Equations not containing x directly 27 
 
 CHAPTER IX. 
 
 SIMULTANEOUS DIFFERENTIAL EQUATIONS. 
 
 Simultaneous equations of first order 28 
 
 Simultaneous equations of higher orders 30 
 
 CHAPTER X. 
 Greometrical applications 32 
 
 Answers to Examples 35 
 
EXAMPLES OF DIFFERENTIAL EQUATIONS. 
 
 o^t^o 
 
 CHAPTER L 
 
 DEFINITIONS. DERIVATION OF THE DIFFERENTIAL EQUA- 
 TION FROM 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 coeffi- 
 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. 
 
2 DERIVATION 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) -{-x = y + c. 
 
 2. (l+a^)(H-2/')=caj2. 
 
 3. cosy=ccosx, 
 
 4. y=z ce-**"'^* + tan-^aj — 1. 
 
 5. y={cx-\-logx-\'l)~\ 
 
 6. y = cx-\-c^c^, 
 
 7. {y-\-cy = 4:ax. 
 
 8. y^s'm^x + 2cy-{-c^ = 0. . 
 
 9. e2^ + 2ca;e^H-c2 = 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, Ci and Cg being 
 the arbitrary constants : 
 
 1. 
 
 2/ = Ci cos (ax+Cj). 
 
 2. 
 
 y = cie" + c^e-". 
 
 3. 
 
 y=(ci + C2x)e'". 
 
 4. 
 
 y = C^ + ^- 
 
 6. 
 
 , cosaa; 
 
 ?/ = Ci sm nx -f Co cos nx + — - 
 
 n^ — or 
 
DERIVATION OF THE DIFFERENTIAL EQUATION. 3 
 
 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. y = Cie^'-]-C2e~^'-\-Cze'. 
 
 2. 2/^* = Cie^'^ + C2sina;V2 + C3COsajV2. 
 
 3. y = U 4- C2X + —je' + C3. 
 
 Form the differential equations of the fourth order of which 
 the following are the complete primitives : 
 
 4. y= (ci + Cgi^-F 03x2)6' + C4. 
 
 6. oi:^ + a^y = Ci e"* + Cg e"*** + c^ sin ax + c^ cos ax. 
 
CHAPTER II. 
 
 DIFFERENTIAL EQUATIONS OF THE FIRST ORDER AND 
 FIRST DEGREE BETWEEN TWO VARIABLES. 
 
 General Form, Mdx + Ndy = 0, 
 where Jf, JSf^ are each functions of x and y, 
 
 8. Form, XYdx + X'Y'dy = 0, 
 
 where X, X', are functions of x alone, and F, F', functions of 
 y alone. 
 
 Divide so as to separate the variables, and integrate each 
 part separately. 
 
 9. Solve the following equations : 
 
 1. {1 -\-x)ydx-{- (l—y)xdy = 0, 
 
 2. (x' - yx") ^ + 2/' + xy' = 0. 
 
 dx 
 
 3. 
 
 dy^_±+jf_, 
 dx {l-\-x'^)xy 
 
 4. a[xJ--\-2y\ = xyJ-' 
 dx J ctx 
 
 5. {l+f)dx=^(y+^l+f){l+xy2dy. 
 
 6. sinaJCOS2/cZa;=cosa;sin2/(^2/- 
 
 7. sec^ajtan2/c?aj + sec^?/taniC(^2/== ^• 
 
 8. sec^a;tan2/d2/H-sec^2/^^^^^^==^' 
 
 9^ dy I 1+y + y' ^Q^ 
 da; 1 + a: + a;2 
 
HOMOGENEOUS EQUATIONS. 6 
 
 10. Homogeneous equations. 
 
 Substitute y = vx\ in the resulting equation between v and a?, 
 the variables can be separated. (See Art. 8.) . 
 
 11. Solve the following equations : 
 
 1. (y — x)dy-j-ydx = 0. 
 
 2. (2^xy — x)dy -{-ydx = 0. 
 
 3. 2/^ + ^— = ^2/—* *' 
 
 dx dx 
 
 4. x^^y+'y/x' + y^ 
 
 dx 
 
 5. xGos^'—=zyeos^ — x. 
 
 X dx X 
 
 6. (8y + 10x)dx+ {oy-i-Jx)dy = 0. 
 
 7. (x + y)-^ = y-x, 
 
 8. xcos^ (ydx-i-xdy) = y sin^ (x dy ^ ydx). 
 
 X X 
 
 9. x-\-y-^ = my. 
 
 dx 
 
 (1), m<2; (2),m = 2; (3), m>2. 
 
 10. [(aJ^ — y^) sma + 2xy cosa — y^x' + 2/T ^ 
 = 2 0^2/ sin a — (a?^ — 2/^) cosa + x -y/x^ + 2/^. 
 
 12. Form, 
 
 (aa; + 6y + c)di»+ (a'a7 4- ?>'2/ + ^0^^ = ^- 
 
 Substitute a; = a;' + a, 2/ = 2/' + i^? 
 
 and determine the constants a, ^, so that the new equation be- 
 tween x' and y' may be homogeneous. (See Art. 10.) 
 
6 LINEAK EQUATIONS OF FIKST ORDER. 
 
 This method fails when — = — . In this case put ax-\-hy = z^ 
 
 and obtain a new equation between x and z or between y and z ; 
 the variables can then be separated. 
 
 13. Solve the following equations : 
 
 1. {3y — 7x-\-7)dx-{- {7y — 3x-{-S)dy = 0. 
 
 2. Ux-\-2y-l)^ + 2x-\-y-{-l=0. 
 
 dx 
 
 3. 
 
 dy _ 7y-\- x-{-2 ^ 
 dx' 3 a; + 5 2/ + 6 
 
 4. {2y + x-\- 1) dx = (2a; + 4?/ + 3)dy. 
 
 5. 2x-y + l-^(x-hy -2)^=^0. 
 
 14. Linear Form, -^ + Py = Q^ 
 
 dx 
 
 where P, Q, are independent of y. 
 
 Solution, y = e'-^^'^y i Qe'^^dx -f cj- 
 
 15. Solve the following equations : 
 
 1. x-^ — ay = x-\-\. 
 
 dx 
 
 2. X {\ ^ x^)dy -{- {2x^ — l)ydx = a:x?dx. 
 
 2^. 
 
 3. (1 - x'Y^ + 2/ Vl - ic' = a; + Vl - a^. 
 
 4. -^ +'ycosa; = -sin2a;. 
 dx ^ 2 
 
 6. (1 + y'^)dx = (tan-^2/ — ^) ^V- 
 
EXTENSION OF LINEAR FORM. 7 
 
 \ dxj 
 
 7. (l+x^)dy-{-fxy--\dx = 0. 
 
 8. ^^y^ = ^^, 
 
 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 —0?)-^ — xy=zax7f. 
 
 dx 
 
 2. 32/2^-a2/3 = a;-f 1. 
 
 dx 
 
 3. ^=2xy(ax'f-^l). 
 
 4. ^(x^f-i.xy)=l. 
 
 6. — + 2/ cos a? = 2/" sin 2 a?. 
 dx 
 
 6. (y\ogx-~l)ydx = xdy. 
 
 7. aa^y^'dy -^-ydx = 2xdy, 
 
 8. y — cosaj-^ =2/^cosir (1 — sina;). 
 
 dx 
 
 9« y-^-\-^y^=^cL008x, 
 dx 
 
CHAPTER III. 
 
 EXACT DIFFERENTIAL EQUATIONS AND INTEGRATING 
 FACTORS. 
 
 18. Mdx + Ndy is an exact differential when 
 
 <1) 
 
 dM^ dJSr 
 dy dx 
 
 The solution of 
 
 Mdx-\' Ndy = 0, in this case is 
 
 Cmdx + ((n- — CMdx\dy = c, 
 
 or CNdy + (i^- — , f^dy\x = c. 
 
 In integrating with respect to a?, y is regarded as constant, 
 and in integrating with respect to 2/, x is regarded as constant. 
 
 19. Solve the following equations after applying the condi- 
 tion (1) for an exact differential : 
 
 1. (aj3 -f- 3 xy^) dx + (^/^ -f- 3 x^y) dy = 0. 
 
 2.. (x^ --ixy — 2y^)dx -f (y^ — 4xy — 2x^)dy = 0. 
 
 3. fl^^\dx-^^-ldy=:0. 
 
 . 2xdx , /I 3a^\,, ^ 
 
EXACT DIFFERENTIAL EQUATIONS. 9 
 
 6. _i^_ + ^l ^ '^^ = 0. 
 
 V?T? \ -Va^-j-yV y 
 
 7. (x + ^ Va; + (^2/ ^dy = 0. 
 
 - - / \ 
 
 8. {l'^e~^)dx-\-e~^ll--]dy = 0, 
 
 9. e(x^-\-y^-+'2x)dx-\-2ye''dy=0. 
 
 10. (m da? + ^ ^2/) sin (wa; + n?/) = (n dx-\-mdy) cos (riaj-f-m^/). 
 
 11. xdx + ydy ydx — xdy ^^ 
 Vl + ar^ -f 2/' a^ + 2/' 
 
 -o of dy — ayx^'^ dx , 
 
 (1), ^>0; (2), ^<0 and =-A;; (3), ^ = 0; 
 (4), a = 0; (5), 5 = 0. 
 
 20. When Mdx-^Ndy 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 = 0, but in that 
 case the solution is ?/ = ex, 
 
 22. Solve the following equations by means of an integrating 
 factor : 
 
 1. (a;^ -\-2xy — y^) dx =^(x^ ^2xy^ y^) dy. 
 X y \y ^J ' 
 
10 INTEGRATING FACTORS. 
 
 4. a^dx-\- {Sx^y-\-2f)dy = 0. 
 
 5. {x-\/x^ + y^ — ^) dy -\- (xy — y -y/x^ + y^) dx = 
 (See Art. 11 for other examples.) 
 
 23. Form, fi{xy)ydx-\-f2{xy)xdy = 0. 
 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 : 
 
 "ps-^l. (1 +xy)ydx-{'(l —Qcy)xdy = 0, 
 
 2. (x^y^-^xy)ydx-{- (x^y^ — l)xdy = 0. 
 
 3. (x^y^ -f 1) {xdy-{- ydx) + (x^y^ + xy) (ydx — xdy) = 0. 
 
 4. (^/xy — l)xdy — (-y/xy -\-l)ydx=0. 
 ^' {y -\-y^/xy)dx-\- {x-\-x^/xy)dy — 0. 
 
 6. e^^ix^y^ -f- xy) (xdy + ydx) -{-ydx — xdy = 0, 
 
 7. xy [1 + cot (xy)'](xdy -{-ydx) + ajd^/ — ydx = 0. 
 
 dM_dN 
 
 25. When ^L^ = ^(^), 
 
 then e*' * * is an integrating factor. 
 
INTEGRATING FACTORS. 11 
 
 dN dM 
 Or, when dx ^dy ^^^y^^ 
 
 then e-^ ^ ^ is an integrating factor. 
 
 26. Solve the following equations by means of an integrat- 
 ing factor : 
 
 1. (aj2 ^ 2/H '^x)dx + 2ydy = 0. 
 
 3. 
 
 (SQ^-y')^ = 2xy. 
 
 dy^ x^-h y\ 
 dx 2xy 
 
 4. [(1 - 2/) Vl -:^-xy'\dx+ [1 - x^ -- x VH^''] dy = , 
 
 5. (cosaj+22/sec2/sec2 2a;)da;-f (tan2a;sec2/— sina7tan2/)(^^=0. 
 
 6. sin(3 X - 2y)(2 dx — dy) + sin {x - 2y)dy = 0. 
 
 7. The Linear Equation 
 
 dx 
 where P and Q are independent of y. 
 
 i+^»=«- 
 
CHAPTER IV. 
 
 DIFFERENTIAL EQUATIONS OF THE FIEST ORDER AND 
 DEGREE CONTAINING THREE VARIABLES. 
 
 General form, Pdx -f Qdy -f- Rdz = 0, 
 where P, Q, i^, are each functions of a?, 2/, 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 : 
 
 \dz dy J \ dx dz J \dy dx J 
 
 The solution may then be effected b^' first solving the equa- 
 tion with one of the parts Pdx^ Qdy, Rdz, omitted, regarding 
 X, y, z, respectively, constant. 
 
 Omitting Rdz, for example, we solve Pdx -\- Qdy = 0, re- 
 garding z constant, and introducing instead of an arbitrary 
 constant of integration, Z, an undetermined function of 2;, 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. ^^ I ^^y I ^^ =0. 
 
 x — a y — b z — c 
 2. (ic — 3 2/ — z)dx -\-{2y — '^x)dy-{-{z — x)dz = 0. 
 
EQUATIONS CONTAINING THRE VARIABLES. 13 
 
 3. (y + z)dx-]-(z-h x)dy -\-{x-\- y)d^ 0. 
 
 4. yzdx-{-zxdy -{-xydz = 0. 
 
 5. (y + z)dx-\-dy -{'dz = 0. 
 
 6 . ay^z"^ dx -\- bz- x^ dy -{- cay^ y^ dz = 0. 
 
 7 . zydx = zx dy -f y^ dz. 
 
 8. (ydx-{-xdy)(a-\-z) = xydz, 
 
 9. (y -{-aydx-\-zdy = (y -\-a)dz, 
 
 10. (y^ + 2/^) do; 4- (xz + 2;^) d?/ + (/ — a;j d^: = 0. 
 
 11. (2x^ + 2xy -\- 2 xz^ -\- l)dx -\- dy + 2dz = 0. 
 
 12. {x^y — y^—y^z)dx-\' (xy^—x^z—x^)u+ {xy^-{-x^y)dz = 0. 
 
CHAPTER V. 
 
 DIFFERENTIAL EQUATIONS OF THE FIKST ORDER, OF A 
 DEGREE ABOVE THE FIRST. 
 
 In what follows, p denotes -^• 
 
 dx 
 
 29. When 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. i)2_5^^e^0. 
 
 2. a?p^-a' = Q. 
 
 3. a;p2_a = 0. 
 
 4. xp^ = 1 — cc. 
 
 6. x^p^ + ^xyp + ^y'^^O. 
 
 7. p^ -\-2 xp'^ — y'^p^ — 2 xy^p = 0. 
 
 8. p^ —(^ -^xy-\- y^)p^ + {x^y -\- ^y'^ -f xf^p — x^y^ = 0. 
 
 9. p^ + 2py cot a; = y^* 
 
 31. When the equation can he solved loith respect to y. 
 Differentiate, regarding p variable as well as x and y^ and 
 
EQUATIONS SOLVABLE WITH RESPECT TO X OR 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 j> between its primitive and the given equation. 
 
 32. Solve the 
 
 following equations : 
 
 1. 
 
 x — yp = ap'. 
 
 2. 
 
 y = xf-\- 2p. 
 
 3. 
 
 {x + ypy = a\\+p') 
 
 4. 
 
 y = xp+p-p\ 
 
 5. 
 
 (y-apy=^l+p\ 
 
 6. 
 
 y = ap + hp^. 
 
 7. 
 
 »? + y=p'^. 
 
 8. 
 
 f=o?(\+p^). 
 
 9. 
 
 y=p^ + 2pK 
 
 33. When the equation can be solved with respect to x. 
 Differentiate, regarding p variable as well as x and ?/, and 
 
 substitute for dx, _Z. There will result a differential equation 
 p 
 
 of the first degree between y and j9. Solve this equation, and 
 
 eliminate p between its primitive and the given equation. 
 
 :. Solve the following equations : 
 
 
 1. 
 
 p^y + 2px = 2/. 
 
 
 2. 
 
 x=p + logp. 
 
 
 3. 
 
 p\x^-{-2ax) = a\ 
 
 
 4. 
 
 x'p^=l-{-p\ 
 
 
 5. 
 
 (x — apy=l+p^; 
 
 also when a= 1. 
 
 6. 
 
 x = ap-{- bp^. 
 
 
 7. 
 
 my - nxp = yp\ 
 
 
16 HOMOGENEOUS EQUATIONS. — CLAIRAUT's FORM. 
 
 35. When the equation is homogeneous with respect to x 
 and y. 
 
 Substitute y = vx. If the resulting equation between p and 
 V can be solved with respect to v^ the given equation comes 
 under Art. 31 or Art. 33. 
 
 But if we can solve with respect to p, substitute forp, v + a;—, 
 
 dx 
 
 and there will result a differentig-l equation of the first degree 
 between v and x, 
 
 36. Solve the following equations : 
 
 1. xy\p^-{-2) =2py^-\-i^. 
 
 2. {2p-^l)xiy=z a; V + ^V^- 
 
 3. 4:X^ = 3{Sy—px){y-{-px). 
 
 4. ds = (^ydx-\- (—) dy, where ds = Vl -i-p^ • dx. 
 
 5. (nx+pyy = {l-^p^){y^-{-nx^). 
 
 37. Clair aufs Form^ 
 
 y=px+f(p). 
 
 The solution is immediately obtained by substituting p=.c. 
 
 38. Solve the following equations : 
 
 1. y=px + -' 
 
 2. y =px -\- 2^ — p^* 
 
 3. y' -2pxy -1 =zp\l -x"). 
 
 4. y = 2p)x-{- y^py^' ^^^^ if = 2/'« 
 
 5. ayp^ -\-{2x — b)p = y. Put y^ = y\ 
 
 6. x^{y —px) = yp^. Put y^ = y\ x^ — x\ 
 
 7. e'^=^(2) — 1)+F^e^'' = 0. Pute'^ = a;', 6^ = 2/'. 
 
 8. (paj — y) (j9y + a;) = ^^jp. Put y^ = y\ a^ = »'. 
 
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 
 
 itive, the result will be a singular solution, provided it satisfies 
 
 the given differential equation. 
 
 41. Let /(cc, 2/, p) = be the given differential equation. 
 By differentiating, regarding p as the only variable, obtain 
 
 S. = 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=rpx-\-—' 
 
 2. y^'-2xyp-\-(l+x^)p^=l. 
 
SINGULAR SOLUTIONS. 
 
 3. y — 4a:2/i)+8i/2 = 0. Futy = z\ 
 
 4. y = (x-l)p-^p\ 
 
 5. y(l+p^) = 2xp. 
 
 6. x^p^ '-2(xy^2)p-^y^ = 0. 
 
 7. (y — xp) (mp — n)= mnp. 
 
DIFFERENTIAL EQUATIONS OF AN ORDER 
 HIGHER THAN THE FIRST. 
 
 CHAPTER VII. 
 
 LIKEAR DIFFERENTIAL EQUATIONS. 
 General Form, 
 
 the coefficients Xi, Xg, ...X„ and X being functions of x alone 
 or constants. 
 
 43. Linear equations with constant coefficients and second 
 member zero msiy be solved as follows : 
 Substitute in the given equation, 
 
 — izsm**, ^ = 7^1** S -^ = m, y = w/=l. 
 
 dx*" dx"-^ 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^r^ ; for each pair of imaginary roots 
 a ± 6V— 1, a term e''''{Asmbx -\- Bcosbx) ; each of the coeffi- 
 cients A^ -B, (7, being an arbitrary constant if the corresponding 
 root occur only once, but a polynomial Ci -|- Cgo; + Cso;^ • • • 4- c^ic''"^, 
 if the root occur r times. 
 
20 LINEAR EQUATIONS. 
 
 44. Roots of auxiliary equation^ real and unequal. 
 Solve the following equations : 
 
 dx^ dx 
 
 3. a^=^. 
 doc^ dx 
 
 4. g(ty+y]=io'M. 
 
 \ dx^ J dx 
 
 c d^v , . dy 
 dor dx 
 
 6. 
 
 ..(,+ii)=<„.+^,|. 
 
 7. ^ = 4^. 
 
 g^ d^y^d^y ^ ^dy 
 dx^ dx^ dx 
 
 dx^ dx 
 
 11. ^ « 2 (a^ + 62) ^ + (a^ - by^ = 0. 
 cZar . dor dx 
 
 45. Boots of auxiliary equation unequal^ hut not all real. 
 Solve the following equations : 
 
 1. g+, = o. 
 
 da? dx 
 
LINEAR EQUATIONS. 21 
 
 3. g_2a^ + 6> = 0. 
 dur ax 
 
 (1), when a>b; (2), when a<b. 
 
 4. ^-iab^+{a' + b'ry = 0. 
 
 5. g-21ogag+[l + (log«)T2/=0. 
 
 6. ^ + 2^ = 0. 
 
 7. ^ = 2/. 
 
 8. i^ + ^ + ^ = 3y. 
 dai? dm? dx 
 
 10. 
 
 
 da;* d»^ 
 
 11. ^ + ia*y = 0. 
 
 46. Auxiliary equation containing equal roots. 
 Solve the following equations : 
 
 1. ^-.2a^ + a22/ = 0. 
 dx^ dx 
 
22 LINEAR EQUATIONS. 
 
 3. ^ = 4^. 
 
 diK^ dic^ die 
 
 6. ^ + 2n2|§ + ^^2/ = 0. • 
 dx^ dor 
 
 dx* do;^ dic^ da; 
 
 8. ^_-4^+14^-20^+252/=0, the first 
 
 da?'* dar dar dx 
 
 member of auxiliary equation being a perfect square. 
 
 9. ^+^!:!^=o. 
 
 dx"" dx*"-^ 
 
 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 7ith order, this value of y will con- 
 tain n arbitrary constants. Derive from it the successive 
 
 dv d^v d"'~^v 
 
 differential coefficients, -^, — ^, • ^ ; then differentiate the 
 
 dx dx^ daf-^ 
 
 values of v? — » — f ? r-? regarding the arbitrary constants 
 
 daj dar dx""'^ 
 
 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 : 
 
 1. ^_7^ + 122/ = a;. 
 dx^ dx 
 
 2. ^-_2^ + 2^-2^ + 2/ = a. 
 dx'^ dx^ dx^ dx 
 
 3. ^^a'y^x+l. 
 dx" ^ 
 
 4. ^_2^ + ^ = e«. 
 dx^ dx^ dx 
 
 5. g-a^^ = a^. 
 
 d^xf 
 
 7. — -^ — 2a-^ -f- a?y = e' ; also when a = 1. 
 dx^ dx 
 
 8. — f 4- n^y = cos ax ; also when a — n. 
 dor 
 
 ^' ^-5^ + 6^ = 6"*; also when n = 2, or n = 3. 
 doer dx 
 
 11. ^^^^^20y = x'^'. 
 dxr dx 
 
 12. — f -\-4y = x sin^ic. 
 dor 
 
24 LINEAR EQUATIONS. 
 
 13. — f + 2— |-f-i/ = a^cosai»; alsowhena=l. 
 
 14. —J — 2-^ + 42/ = e*cosi». 
 
 (XX/ (XX 
 
 49. Linear equations of tlie form 
 
 where -4i, ^2) '"-^n ^-re constants, and X a function of a? alone. 
 
 Put o. +bx = e\ and change the independent variable from 
 X to t. The new differential equation between y and t will be 
 linear with constant coefficients, and may be solved by Art. 47. 
 
 50. Solve the following equations : 
 
 dxr dx 
 2. {X + ay^ _ 4 (oj + a) ^ + 6y = x. 
 
 3- (« + ^^y^ + b{a + bx)^ + hhj = 0. 
 dar do? 
 
 4. a;2^ _ aj^ + 22/ = x\ogx, 
 da? dx 
 
 6. i6(«,+ l)^^ + 96(a. + irg+104(..+ irg 
 
 + 8(a;+l)^ + 2/ = a^ + 4a! + 3. 
 da; 
 
 7. a,.^ + 6af'f| + 9a;^g + 3a;^ + 2/ = (l+loga,r. 
 
 do;* dar dar aa? 
 
 8. x"^ - (2m - 1)0?^ + W + n^)y = n^a^-loga?. 
 
 dx^ dx 
 
CHAPTER VIII. 
 
 SOME SPECIAL FORMS OF DIFFERENTIAL EQUATIONS OF 
 HIGHER ORDERS. 
 
 51. Form, -— ^ = X, where X is a function of x alone. 
 
 The expression for y is found by integrating X successively 
 71 times with regard to x. Or solve by Art. 47. 
 
 52. Solve the following equations : 
 
 1. x^=i. 
 
 2. 
 
 d'y_ 
 
 dx'^ (ic-j-a)^ 
 
 3. ^=af. 
 dx*^ 
 
 4. 
 
 d'y 
 
 — ^ = oleosa;. 
 
 dx'^ 
 
 5. e"" — ^ -f- 4 cos i» = 0. 
 
 dx' 
 
 6. ^ = xe', 
 dx"" 
 
 7. — ~ = sin^ic. 
 da^ 
 
 (jpy 
 
 53. Form, — ^ = Y^ where Y' is a function of y only. 
 dor 
 
 Multiplying both members by 2-^, and integrating, we have 
 
 (^=2 Crdy + c^, Therefore x=: f ^ 
 
 + C2. 
 
26 EQUATIONS NOT CONTAINING y. 
 
 54. Solve the following equations : 
 
 2. ^ = -aV 
 
 3. ^^^ = a. 
 
 4. tl^^e^y. 
 
 5. 
 
 do^ -yjay 
 
 55. Equations not containing y directly. 
 
 By assuming the differential coefficient of the lowest order in 
 the given equation equal to z^ and consequently the other differ- 
 ential coefficients equal to the successive differential coefficients 
 of z 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. aj^ + ^ = 0. 
 
 dx^ dx 
 
 2. ^ = a^J^h' 
 da^ 
 
 dy 
 dx 
 
 doer \dxrj 
 
 «■ "•©■--(IT 
 
 5. 
 6. 
 
 
EQUATIONS NOT CONTAINING X. 
 
 27 
 
 8. 
 
 10. 
 
 doer dx 
 
 da^ dx" [dx^J 
 
 dx^ dx" \ ' doj^yL V^^y_ 
 dx^ \dxj 
 
 57. Equations not containing x directly. 
 By assuming -^=.z^ and consequently 
 
 ^^z—, ^^z'^^^ + zf^^Y, etc., 
 
 changing the independent variable from x 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. 
 2. 
 3. 
 4. 
 5. 
 6. 
 
 • ^^^y) -j3+(^ + ^^Ey)(-~^ 
 
 W'- 
 
 dxr \dxj 
 2/(1 
 
 dor L\^^/ \darj J \dx_ 
 
 \dx) ''da? dx •' \\dx) dx" j 
 
 dx"' 
 
CHAPTER IX. 
 
 SIMULTANEOUS DIFFERENTIAL EQUATIONS. 
 
 59. Simultaneous differential equations of the first order. 
 There should be n given equations between n-{-\ 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 differ- 
 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 71 — 1 times 
 successively, we shall have in all n^ equations, which are just 
 sufficient for the elimination of n — 1 variables, together with 
 their n(n — l) differential coeflScients. Shorter processes for 
 the elimination will frequently suggest themselves in special 
 cases. 
 
 60. Solve the following simultaneous equations : 
 
 1. 
 
 !+*'+!-»• 
 
 
 | + 3,-« = 0. 
 
 2. - 
 
 ax- dy __ dz 
 
 ^y + 4tz 2y-\-bz 
 
 3. - 
 
 ^^ _ -^y ^dt. 
 
 y — 7x 2x-\-5y 
 
SIMULTANEOUS EQUATIONS OP THE FIKST OBDEE. 29 
 
 ax 
 
 dx 
 
 + a2!=0. 
 
 5. 
 
 dt 
 
 dy 
 dt 
 
 ^3y — x=e^' 
 
 dx 
 
 ____dy_ 
 
 2y — ox + e* x — 6y-\-e^* 
 
 = dt. 
 
 dt dt 
 
 3— + 7^ + 34 a; + 382/ = e*. 
 dt dt 
 
 dt dt 
 
 S^^7^-\-Sx-{-24.y = e'K 
 dt dt •" 
 
 10. 
 
 4— + 9^ + 2a; + 3l2/ = eS 
 dt dt 
 
 3— 4-7^ + a; + 242/ = 3. 
 ^ dt dt 
 
 dt t^ ^^ 
 dt t^ ^^ 
 
 11. 
 
 i; 
 
 tdx= {t — 2x)dt, 
 dy=z{tx-\'ty + 2x — t) dt. 
 
30 SIMULTANEOUS EQUATIONS OF HIGHER ORDERS. 
 
 12. 
 
 ( dx 
 dt ^ 
 
 -^ = lz'- nx, 
 dt 
 
 = mx — ly. 
 
 13. 
 
 f dx 
 lt— = mn{y-z), 
 at 
 
 7nt~^ =7il (z — x). 
 dt ^ ^' 
 
 nt — = Im (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 : 
 1. 
 
 dv 
 
 —4^ — n^x = 0. 
 dt^ 
 
 d^x 
 
 1 
 
 df" 
 d'y 
 
 dt:- 
 
 ■Sx — 4:y-\-S = 0, 
 
 ^2 +x + y + 5=:0. 
 
SIMULTANEOUS EQUATIONS OF HIGHEB OBDEES. 31 
 
 — -3a;- 42/ + 3 = 0, 
 
 ^+a;-8w + 5 = 0. 
 df ^ 
 
 4. 
 
 (d'x , .,2 
 
 df 
 
 + n'y = % 
 
 — n^x = 0. 
 
 5. 
 
 r2^-^-4y = 2a;, 
 daj^ do; 
 
 2^ + 4— -32^ = 0. 
 do; dx 
 
 
 dx' 
 
 " da^ ' do^ 
 dx^ dx^ 
 
 
CHAPTER X. 
 
 GEOMETKICAL EXAMPLES. 
 
 63. Expressions involved in the examples, p representing 
 
 -^, and q representing — |. 
 dx dar 
 
 y 
 
 Subtangent=-. Subnormal =py. 
 
 . ds 
 
 Normal = 2/ Vl 4-i>^. "T~=VlH-p^. 
 
 y 
 
 Intercept of tangent on axis oi X=x 
 
 Intercept of tangent on axis of Y=y—px. 
 
 Radius of curvature = q: -^ — JULJ—, 
 
 Q 
 
 1. Find the curve whose subtangent varies as (is n times) the 
 
 abscissa. 
 
 2. 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 = mx^, 
 
 5. Find the curve for w^hich 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 : 
 
 6. y = mx, m being the variable parameter. 
 
GEOMETRICAL EXAMPLES. 33 
 
 7. 7f = 2aa; — ic^, a being the variable parameter. 
 
 8. y = 4aa7, a being the variable parameter. 
 d, xy = k^^ k being the variable parameter. 
 
 10. f- ^ =z 1 , h being the variable parameter. 
 
 11. x^ -\- m^ if = m^ a^ ^ a being the variable parameter. 
 
 12. h — = li ^ being the variable parameter. 
 
 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 Fis constant and equal to a, 
 
 14. Find the curve such that the part of the tangent betweei\ 
 
 the axes of X and Fis constant and equal to a. 
 
 16. 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. 
 
AlsTSWERS. 
 
 Art. 3. (p = ^ 
 
 dx 
 
 ) 
 
 1. y{l-{-x)-\-px{\-y) = 0, 6. y=px-j-2^-p\ 
 
 2. {x^-\-l)pxy = y^ + l. 7. xp^ = a, 
 
 3. tan a; = p tan 2/. 8. p2 + 2p2/cota; = 2/^. 
 
 4. (l4-a;2)p+2/ = tan-^a;. 9. x'p^ = l-\-p\ 
 
 5. (2/logic— 1) ?/=pic. 
 
 Art. 5. 
 
 1. ^4.a^2/ = 0. 4. a.^^_a;^=3y. 
 dx^ dor dx 
 
 2. — ^ — a'^y = 0, 5. — | + 7i^2/ = cos aa;. 
 daj2 dxr 
 
 3. ^-2a^ + a^2/ = 0. 
 dor dx 
 
 Art. 7. 
 dx^ dx ' * dx^ dx^ dx^ dx 
 
 dxr dor dx dx* 
 
 3. ^_2^ + ^ = e^ 
 dx^ dx^ dx 
 
 Art. 9. 
 
 1 . log (xy) -^x — y^c. t^ . (1 + x^) (1 + 2/^) = cx^. 
 
 2. ?Jl^4.w^ = c. 4- xhj = ce^. 
 
 xy ^ ^x 
 
36 
 
 ANSWERS. 
 
 5. log[(2/ + Vl+2/')Vl+/] 
 
 + c. 
 
 6. cosy = cGosx. 
 
 7. tana;tan^ = c. 
 
 1. y = ce~K 
 
 2. yz=ce-yl, 
 
 3. y = ce^, 
 
 4. x^ = c^ + 2cy.Xj^\ 
 
 8. sin^ X+ sin^y = c. 
 
 9. xy—l = c{x-{-y + l). 
 
 Art. 11. 
 
 5— sin? 
 . X = Ce a; . 
 
 6. {y + xy(y-{^2xy = c. 
 
 7. log (aj2 + /) = 2 tan-i - + c. 
 
 2/ 
 
 8. xycos'-=c. 
 
 X 
 
 9. (1) , log (a;2 — mxy + 2/^) + 
 
 (2), x — y = ce^~ 
 
 2 m 
 
 rtaii 
 
 _i 2v — mx 
 
 X -y/4 _ -^2 
 
 (3), 
 
 (2y — mx 4- a?V?7i^ — 4)" 
 
 (22/ - mx - a;Vm2-4)'"+^"^'"' 
 10. 2/sina — a;cosa + Va^ + 2/^ = c (a^4-2/^)' 
 
 Art. 13. 
 
 1. (^y — x-}-iy(y + x-iy = c. 
 
 2. x + 2y-{-log{2x + y—l)=c. 
 
 3. x-\-5y + 2=:c(x — y-\-2y. 
 
 4. 4aj — 82/ = log(4aj4-82/ + 5) +c. 
 
 6. log[2(3a;-l)^+(3y-5)^]-V2taD-^ '^^^^^~-^^ =c. 
 
 3y — 5 
 
1. y = cx'' + 
 
 1—a a 
 
 ANSWERS. 
 Art. 15. 
 3. 2/ = 
 
 37 
 
 ^ __4-ce^^. 
 
 2. y = ax-{-cx-\/l —x^. 
 
 4. 2/ = sina;-l+ce-«^°". 
 
 7. 2/vrT^ = iog^^i-±^^^+<^- 
 
 Art. 17 
 
 3 
 
 1. y = {c-y/l-x'-a)-\ 
 x-]-l 1 
 
 2/= ' 
 
 ce2.^4.^(2a^ + l)T^. 
 
 2. 2/^ = ce^ 
 
 4. ic = 
 
 5^ 2/""'^^ = ce^""^^"""' + 2sina; + 
 
 2 
 
 6. y = {cx + \ogx + l) \ 
 
 8. 2/ = 
 
 _(n±2)f o - tana^ + seco; 
 
 a^r+' + c \ sinx + c 
 
 9. (4&2 + 1) 2/2 = 2a(smaj 4- 2&C0SX) + ce^^^^ 
 
 Art. 19. 
 
 1. x^-|-6a^2/' + 2/' = c- 
 
 2. a^-Qx'y-exy' + f^c. Q. y' = c'-2cx. 
 
 3. x^ — y^=cx. 
 
 X 
 
 5. a^ + 2/^ + 2tan-i| = c. 
 
 7. a^ + 2/^ + 2sin-i-==c. 
 
88 ANSWERS. 
 
 9. e(p^^y^)^c, 
 
 10. cos {mx + ny) + sin {nx =f my) = c. 
 
 11. VTT^T? + tan-i2=c. 
 
 12. (1), iog-:5!^^i±r^=2W%_^^^ 
 
 (2), tan-i^ + ^!V6^ = e. 
 
 (3), a^(^l_ lUe. 
 Va hy) 
 
 (4), ^^±lVf^,^.v^.^^ 
 
 a ^a;« 
 
 Art. 22. 
 
 1. x' + y^ = c{x + y). 4. aj2 + 22/2=cV?T?. 
 
 2. aj2__^2^^^^^^ ^ y = cx. 
 
 3. 2/ = cflj. 
 
 Art. 24. 
 
 1 
 
 1. x=icye^. 4. -4= = log — 
 
 2- ^ = ce^^ 5. xy=^c, 
 
 3- ^^- — = logc/. 6. a;2/e^ = log^. 
 
 a; 
 
 ca? 
 
 7. «2/ + log sin (xy) = loff — 
 
 y 
 
 Art. 26. 
 
 1. e{^-^^)=.c. 3. a^2_^2^^^^ 
 
 2. d^-.f^of. 4. 2/Vn^ + a:(l-2/)=c. 
 
ANSWERS. 39 
 
 5. sinxcosy-{-ytsin2x = c. 6. siii^a;sin2 (aj — ^/) = c. 
 7. y = e~J''"'ffQe-'''"''dx + c\ 
 
 Art. 28. 
 
 1. {x^a)(y — b){z — c) = c'. 7. z = c^. 
 
 2. a^+22/^— 6a?2/— 2i»2;+2;^=c. 8. cc^/ = <^ (^ + ^) • 
 
 3. yZ'^zx + xy = c. 9. a; = f-c 
 
 2/ + ^ 
 
 4. xyz = c. 10. 2/(a? + 2;) = c(2/ + 2;). 
 
 5. 6=^(2/ + 2;) = c. 11. e^(a; + 2/ + ^0 = c. 
 
 6. ^ + 5 + ? = c'. 12. ^±_%^_±^ = e. 
 x y z X y 
 
 Art. 30. 
 
 1. (i/-2aj + c)(?/-3a; + c) = 0, or {bX'-2y -\- cy=^o?. 
 
 2. (?/ + c)2 = a2(loga;)2. 3. {y-^cy = 4.ax. 
 
 4. (2/ + c)2 = ( Va? - ar^ + sin-^ VS)^ 
 
 5. {xy + c){o^y-Jrc)=0. 
 
 6. (x2-22/ + c)[e^(aj4-2/-l) + c]=0. 
 
 7. (2/ + c)(2/ + ^ + c)(ar2/ + c2/+l) = 0. 
 
 8. (x^-32/ + c)(e2+c?/)(a;i/ + c2/+l) = 0. 
 
 9. 2/2sin2a;4-2c2/ + c2 = 0. 
 
 Art. 32. 
 
 1 . Eliminate p by means of x = — z=- (c -f- a sin~^p) . 
 
 Vi — p- 
 
 2. Eliminate p by means of a;(p — 1)^ = logp^— 2p + c. 
 
 3. Eliminate p by means of a? = — , ( c H f- atan"^» ). 
 
40 ANSWERS. 
 
 4. y = cx-^C'-c^. 
 
 5. a; = alog {ay ± Va' + 2/^—1) + log (y T Va^ + 2/^ — 1) + c. 
 
 6. a;± Vc?+46^ = alog(a± Va2 + 46?/) +c. 
 
 w /« L /-2-^ — \^i7 d=4 Vi«^ + ^ — i:c( Vl7 — 1) 
 
 7. c(22/±a;Va;2 + 2/) = ^ , -. 
 
 y + V?/^ — of 
 
 = (2/2 + a^logca^)2 
 
 8. 2/V/-a^-a^log- ^ 
 
 9. 4(a; + c)3+(x + c)2-182/(a; + c) - 272/'- 4^/ = 0. 
 
 Art. 34. 
 
 1. y^ = 2cx + c^. 
 
 2. 0^ + 1 = ± V22/ + c + log(± V22/ + C-1). 
 
 1 y 
 
 3. (e« — ac)2r=2caje«. 
 
 4. e-^ + 2ca;e^ + c2=0. 
 
 a'' — 1 
 when a=l, 4^/ = a;' — log (ca^). 
 6. 62(62/ + c)2+(6a&aj + a3)(62/ + c)-3a2a;2-16&aj3 = 0. 
 
 7. c(naj2+22/2±ajVnV+4m2/2)»^= [(2m~n)a;± VnV+4m2/']^ 
 
 Art. 36. 
 
 1. (a^ - 2/2 + c) (0^2 - 2/' + c^') = ^• 
 
 3. 3a;* + 6ca^2/ + ^' = ^- 
 
 4. (2/-cc)^^^2^c(V^ + V^)l 
 
 6. x^+2cx*-iy-o27i = 0, where A = ^^- 
 
ANSWERS. 41 
 
 Art. 38. 
 
 2. y z=cx-{-c — c^. 6. 2/^ = cx^ -\- (?, 
 
 3. (2/ — ca;)2= l+c^. 7. e^=ce^ + c"\ 
 
 4. 2/'=cx + — • 8. y^-cx^- ^^'' 
 
 c-hl 
 
 Art. 42. 
 Complete Primitives and Singular Solutions : 
 
 1. 
 
 y = cx + ^, 
 
 C 
 
 
 2/2 = 4 mo;. 
 
 2. 
 
 (y — cxy== 1— c^, 
 
 
 f^^ = \. 
 
 3. 
 
 y = c(x-cy, 
 
 
 y='-\ 
 ^ 27 
 
 4. 
 
 y=ic{x-\)-(?. 
 
 
 42/=:(aj~l)2. 
 
 5. 
 
 2/2_2caj + c2 = 0, 
 
 
 f=^x\ 
 
 6. 
 
 (2/~ca^)2+4c = 0, 
 
 
 xy=l. 
 
 7. 
 
 {y — cic) (mc — n) = mnc^ 
 
 \mj \nj 
 
 
 Art. 
 
 44. 
 
 
 ax hx 
 
 2. 2/ = Cie2* + C2e**. 6. 2/ = Ci^^ + Cae". 
 
 X 
 
 3. 2/ = Cie« + C2. 7. 2/ = Cie2^ + C2e-2*+C3. 
 
 4. y = Cie^'' -\- C2eK 8. 2/ = ^i^^"" + 026-2^ + 03. 
 
 9. y = Cie^''-\-C2e~^''-^Cse', 
 
 10. 2/ = Cie^* + 026"^=" + 636=^^3 + 646-^^^ 
 
 11. 2/ = Oie^"-^^^ + Cge^*-"^^ + C3e^«+^>^ + C4e-(«+^>* + C5. 
 
42 ANSWERS. 
 
 Art. 45. 
 
 1. 2/ = CiSina; + C2COsa?. 
 
 2. 2/ = e^*(ciSin2a?4-C2COs2aj). 
 
 3. When a > 6, 3/ = e«*(cie^v^^ + C2e-^V^^-^') 5 
 
 when a<hj y=e'^(cismx V6^ — o?-\-C2 cos x V^^ — o?) . 
 4.2/ = e^"^* [ci sin (a^ - Z?^) a; + Cg cos (a^ - W) x'] . 
 
 5. 2/ = a*(cisina; + C2C0sa;). 
 
 6. 2/ = Ci sin a; V2 + C2C0sa? V2 + C3. 
 
 -* / a; "v/S a; "v/3\ 
 
 7. 2/ = Cie='4-e ^^sin-^— + C3eos-^j. 
 
 8. 2/6"" = Ci e^* + C2 sin x V2 + C3 cos x V2. 
 
 9. 2/ = Cie* + C2 6~* + C3sina;-f C4eosaj. 
 
 10. y = Cie*^2 _)_ c2e-*^2 + Cssin 2 a: + C4COS 2 aj. 
 
 11 . 2/ = e*" (ci sin ax + Cg cos ax) + e"*** (cg sin ax + C4 cos ax) . 
 
 12. 2/ = Cie* + C2e-* + (0362 + 046 2) sin ^^ 
 
 a;V3 
 
 + (0562 + C6e 2) cos 
 
 13. 2/ = Ci sinoj + C2 cos x-\-e ^ ( C3 sin - + C4 cos - j 
 
 I ^ — 2~i . aJ , a;\ 
 
 + e 2 /c5Sin- + C6Cos-j. 
 
 14. y = Cie="+ 026"*+ Cgsina; + C4COsa;+e^Y^5sin— +C6Cos— ) 
 
 V V2 V2/ 
 
 + e ^^2/'c7sin-^ + C8Cos-^Y 
 \ V2 V2y 
 
 Art. 46. 
 
 1. 2/= (ci + C2aj)e«'. 2. 2/ = Ci + C2a;. 
 
 3. 2/ = Cie*' + C2 + C3a;. 4. 2/ = ^1^-*+ (02 + 033;) e^' 
 
ANSWERS. 48 
 
 6. y = (Cj 4- ^2^) cosnx + (cg + c^x) sinnx, 
 
 8. ?/ = e=^ [ (cj + Cgic) sin 2 aj + (cg + 040?) cos 2 a;] . 
 
 9. y = Ci-\-C2X-{'CsX^ '" +Cn_2X''~^ + c^^iS\nx + c^cosx, 
 
 1. 2/ = Cie3=« + C2e^^-+ 
 
 Art. 48. 
 
 12a;+7 
 
 144 
 
 2. 2/ = CiSina; + C2COsaj + (C3 + C4a;)e'-fa. 
 
 3. 2/ = Cie«' + C2e-«* — ^-i-i. 
 
 0. y = CiS'^' -\- CiB ""' -\-c^^\nax-\-c^Q,osax • 
 
 a* 
 
 0. 2/ = Ci sin Tix + C2 cos no? + ' 
 
 7. 2/ = (Ci + C2a;)e«* + 
 
 
 when a = l, 2/=[^i + ^2^H je^ 
 
 8_ . , , cos aa; 
 . y = Ci sin na; + C2 cos nx -|- — ; 
 
 X sin 7tx 
 when a = 71, 2/ = ClSln^i»^-C2Cos?^a; + - 
 
 9. 2/ = Cie2- + c2e3«-|- 
 
 2ri 
 
 when n = 2, 2/ = (^1 — ^)6^'' + C2 6^*; 
 when 71 = 3 , 2/ = ^i ^^'^ + (^2 + ^0 6^''- 
 10. y^c,e^ + c,e^' + -J;^ (2n-3)e'" 
 
44 ANSWERS. • 
 
 11. y = c,e'' + c,e" + 2^^ + 6a; + 7 ^.^ 
 
 12. y=U-^^^m'ix + (c,-^co^2x + '^. 
 
 13. 3/ = (Ci + C2a;) siiia; + (c3 + C4a;) cosa; 
 
 1 smaj 
 
 
 I cos a:;. 
 ^x\ 
 
 Art. 50. 
 
 a? 
 
 1. 2/ = Cia^+ ^ 
 
 3. 2/ = Ci sin log (a + hx) + Cg cos log (a + 5a3) * 
 
 4. 2/ = ^(CiSiiiloga; + C2COsloga7 + logaj). 
 
 5. 2/ = (2a;-l)[ci + C2(2a!-l)2 +C3(2a;-1) ^]. 
 
 6. y = [., + c. log (X + 1)] V^Tl + '^ + '^ !°^.^^, ^ ^^ 
 
 \lx + 1 
 
 225 
 
 7 . 2/ = (^1 + ^2 log a;) sin log oj + (cg + C4 log a;) cos log x + (log a;) * 
 
 + 21oga;-3. 
 
 8. y = af"(ci sin logx** + Cgcos logo;" + logo;) . 
 
 Art. 52. 
 
 1. 2/ = Ci + CaOJ + Cga^^ + a^logaj. 
 
 2. 2/ = Ci + C2a? + C3a:^ + C4X^ — (aj-f a)^log Vic + a. 
 
ANSWERS. 45 
 
 3. y = Ci + C2X -{- CqX^ " • -{- c,, x^~^ + 
 
 m -\-7i 
 
 4. 2/ = Ci-{-C2X'i-CsX^ -{-c^a^ -{-xcosx — 4 sin a;. 
 
 5. y = Ci + C2X-{-CsxF -{-c^x^ -{-e~'cosx. 
 
 6- 2/ = ^1 + ^2^ + CgOJ^*.* + c„aj"~^ + (a; — n)e'^- 
 
 Art. 54. 
 
 1. ax = \og{y + -\/y^-{-Ci)-^C2, or ?/== c'ie'^* + c'ae""*. 
 
 -1^ 
 
 Ci 
 
 3. (c,x + C2y + a=c,y\ 
 
 2. ax = sin ^^ -f- Cg, or y = Ci sin (ax + C2) . 
 
 Ci 
 
 4. xV2¥=c,log^^g^^^^+"^-^+c. 
 
 5. Sx=2a'(y^ -^2c,)(y^ + c,y +C2. 
 
 Art. 56. 
 
 1. y = Ci\ogx + C2. 
 
 2. 6^2/ = log sec [a&(a; + Cj)] + Cg. 
 
 3. y = '-l^ + xf(cO+C2. 
 
 a 
 
 5. (x + c0^4-(2/ + C2)' = ^'. 
 
 6. 2/ = CiX+ (ci2+l)log(x-Ci) +C2. 
 
 7. 2/ = Cisin~^a: 'r (sin~^a;)^ + C2. 
 
 4 - 
 
 8. y = -—(x + c,^a^)^-{-C2X-\-Cs. 
 
46 ANSWERS. 
 
 9. l2y = w^ + Citv — 6wlogW'j'C2, where w = x + Cs, 
 
 10. 2 2/ V^= w -y/w' + Ci^ + c/log (w; -I- Vti?Tc?) + Cj, 
 where w = x -\- c^. 
 
 Art. 58. 
 
 1. y' = x' + c^x + C2. 4. Cii»=:plog[ci2/+/(ci)]+C2. 
 
 2. log?/-l = . 5. log2/ = Cle^ + C2 6-^ 
 
 ^1 "^ "T" ^2 
 
 3. c^y = C2e''i^ — VT+oFc^K 6. 2/^+^ = Cie''^* + Cg. 
 
 Art. 60. 
 (2x = (2c2 — Ci — C2t)e ^, 
 
 
 
 2 a; = e~^'[ (ci + Cs) sin ^ + (cg — Ci) cos i] , 
 
 g |2aj = e-^'L(Ci + C2 
 1?/ = e-6*(ClSm^^- 
 
 ■ C2 cos ^) 
 '2/ = Cie"* + C2e-' 
 
 az = — nci e"""" + ^Cg e 
 
 
 n^-l 
 
 6. 
 
 ^ ' ^ 36 25 
 
 J/ = - (c, + C2 4- C20e-^' + ^V -• 
 
 27 40' 
 
 y=:c,e''^ + C2e-'^ + l^ + ^. 
 54: 40 
 
ANSWERS. 
 
 47 
 
 , , _6, 29e' , 19^ 56 
 r , .\ -At 49 e2* 31 e' 
 
 36 
 
 2/ = -(Ci + C2 + C20e~*' + 
 
 25 ' 
 19 e^^ lie' 
 
 36 
 
 25 
 
 9. 
 
 10. 
 
 (. . , ,\ _4/ , Ox S \jO 
 
 Cismt + c^cost) e ^^ + __-_, 
 
 Zb 1 / 
 
 2e' 
 
 J = [(^2 - Ci) sin^ - (C2 + Ci) cos^] e"*' "J^'^Jj' 
 
 10 15 
 ^ ' ' 20 15 
 
 11. 
 
 'x = Cit-^+-, 
 
 y^c^e'-c^t 2__. 
 
 12. ^ 2/ = ^1 sin ^t 4- &2 cos A:^ 4- ?>3, 
 2; = Ci sin A:^ + C2 cos A;^ + C3, 
 where A:^ = Z^ + m^ + 7^^ 
 
 The arbitrary constants are connected by the following equa- 
 tions : 
 
 mci — nhi _ nai — hi __ Ibi — maj _ , 
 
 0/2 O2 C2 
 
 « m n 
 
48 ANSWERS. 
 
 {X = ajsin {h\ogt) + agcos {klogt) + ag, 
 2/ = &i sin (/clogO + &2 cos (A:logO + ^s? 
 z=.Ci sin (/clog^) + C2 cos (A:log^) + C3, 
 
 where A;^ = Z^ + m^ + n^. 
 
 The arbitrary constants are connected by the following equa- 
 tions : 
 
 mn (ci — &i) _ 7i?(ai— Ci) __ Im (61 ~ %) _. j^ 
 
 ?ai + m^6i + n^Ci = 0, a^=^ h^ =03. 
 
 1. 
 
 2. 
 
 3. 
 
 Art. 62. 
 
 ' flj = Ci sin nt -f Cg cos n^, 
 
 {:: 
 { 
 
 aj = (ci + C2O e' + (C3 + C4O e-' - 23, 
 
 - 2 2/ = (ci - C2 + C2O e* + (C3 + C4 + c^O^"' — 36. 
 
 €/ . n?5 , ^i\ , -^/ . nt nt\ 
 
 To/ . nt nt \ , -%/ . ?i^ , ^^^^ \ 
 
 r ii/ = (ci + c^x) (f + Scse 
 5. f 
 
 T_?, 
 
 6. i _3^ 1 
 
 U = 2(3c2 — Ci— C2i:c)e=' — Cge ^""3* 
 
 6. y = u + v^ z = — u + v^ 
 
 where ?^ = c^ e^- + (cg e-* + C3 e""') e«, 
 
ANSWERS. 49 
 
 where ii = (ci + C2X + c^e"^^ + c^e-'''^^)e% 
 
 Art. 63. 
 
 1 . a? = C2/**. 
 
 2. 2/^ = 4 aa; + c, a parabola. 
 
 3. ±(a; + c)=log(2/ + V/-l), 
 
 or 2/ = i(e''"^'' + e'^ ''j , a catenary. 
 
 4. 4 m^/ -f- c = 2 mx ■\/4m^a^— 1 + log (2 mo; — V4 m^a^ — 1) . 
 
 5. ±{x-\- c) = alog {y + V^/^ — a^) , 
 
 or ±{x + c)= aXo^y + ^^' ~ ^^ 
 
 a 
 
 from which ^/zn-le^+e "j, a catenary. 
 
 6. a;^ 4- 2/^ = c^, a circle. 
 
 7. ic^ + 2/^ — 2 C2/ = 0, a circle. 
 
 8. 2 0^ + 2/^ = 2 c2, an ellipse. 
 
 9. :k? — y'^z=c^^ an equilateral hyperbola. 
 
 10. 2/^ + ic^ = a21oga;^ + c. 
 
 11 . y = cx^^, 
 
 X^ ^2 
 
 12. — ^ — -^ =1, an ellipse or hyperbola. 
 
 /r — c^& 
 
 13. a;^ + 2/^ = ot^, a parabola. 
 
 14. x^ -\-y^ = a^^ a hypocycloid. 
 
 16. 2xy=^a^^ an equilateral hyperbola. 
 16. c^2/^ — log 2/^ = 4 c(a; + c'). 
 
50 ANSWERS. 
 
 17. Gif" {x-\- c'y= 1, a hyperbola, when n > ; an ellipse 
 
 n 
 
 or hyperbola, when n < 0. 
 
 18. Fu-st, (x + c')2 + 2/2 = C-, a circle. 
 Second, ±(x-i-c') = c log {y + V^/^ — c^) , 
 
 a catenary. 
 
 19. First, x-\-c' = c vers ^ '\/2cy — 2/^, a cycloid 
 
 Second, (a; + c') ^ = 2 ci/ — c^, a parabola. 
 
 or ±(a; + c')-clog^+^^'" 
 
 c 
 
 from which y = -le '^ -{-e " L 
 2\ / 
 
 ?/ 
 
 a ( 
 
MATHEMATICS. 81 
 
 Wentworth's College Algebra. 
 
 By G. A. Wentworth, recently Professor of Mathematics, Phillips 
 Exeter Academy. Half morocco. 500 pages. Mailing price, $1.65 ; for 
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 rPHIS is a text-book for colleges and scientific schools. The 
 
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 Yale University: I find it charac- 
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82 MATHEMATICS. 
 
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MATHBMATICa 83 
 
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84 MATHEMATICS. 
 
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MATHEMATICS. 85 
 
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