Elementary Differential Equations

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Elementary
Dijferential
Equations
Sixth Edition

Earl D. Rainville
Late Professor of Mathematics
University of Michigan

Phillip E. Bedient
Professor of Mathematics
Franklin and Marshall College

Macmillan Publishing Co., Inc.
New York

Collier Macmillan Publishers
London

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Copyright© 1981, Macmillan Publishing Co., Inc.
Printed in the United States of America
All rights reserved. No part of this book may be reproduced

or transmitted in any form or by any means, electronic or
mechanical, including photocopying, recording, or any
information storage and retrieval system, without permission
in writing from the Publisher.
Earlier editions copyright© 1949 and 1952, © 1958, and
copyright© 1964, 1969 and 1974 by Macmillan Publishing
Co., Inc. Some material is from The Laplace Transform:
An Introduction, copyright© 1963 by Earl D. Rainville.
Macmillan Publishing Co., Inc.
866 Third Avenue, New York, New York 10022
Collier Macmillan Canada, Ltd.
Library of Congress Cataloging in Publication Data
Rainville, Earl David,
Elementary differential equations.
Includes index.
1. Differential equations. I. Bedient, Philip
joint author. II. Title.
Edward,
QA371.R29 1980
515.3'5
80-12849
ISBN 0-02-397770-1
Printing: 12345678

Year: 12345678

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Preface

to the Sixth Edition

This new edition of Professor Rainville's book maintains the simple and
direct style of earlier editions and makes some modest changes. The balance
between developing techniques for solving equations and the theory necessary to support those techniques is essentially unchanged. However, the
variety and number of applications has been increased and placed as early
in the text as is feasible.
The material is arranged to permit great flexibility in the choice of topics
for a semester course. Except for Chapters l, 2, 5, 16 through 18, and either
6 and 7 or 11 and 12, any chapter on ordinary differential equations can be
omitted without interfering with the study of later chapters. Parts of chapters
can be omitted in many instances.
For a course that aims at reaching power series as rapidly as is consistent
with some treatment of more elementary methods, a reasonable syllabus
should include Chapters 1 and 2, Chapters 5, 6, 7, 8, parts of Chapters 13
and 15, Chapters 17 and 18. and whatever applications the instructor cares
to insert.

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v


vi

Preface to the Sixth Edition

Chapters 1 through 16 of this book appear separately as A Short Course
in Differential Equations, Sixth Edition. The shorter version is intended for
courses that do not include discussion of infinite series methods.

The author wishes to thank those students and colleagues at Franklin
and Marshall College whose suggestions and support have been most helpful.
Phillip E. Bedient

Lancaster, Pennsylvania

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Contents

1

Definitions, Elimination of Arbitrary Constants
1.
2.
3.
4.

2

Examples of differential equations
Definitions 3
The elimination of arbitrary constants
Families of curves 10

5

Equations of Order One
5.

6.
7.
8.
9.
10.
11.

The isoclines of an equation 16
An existence theorem 19
Separation of variables 20
Homogeneous functions 25
Equations with homogeneous coefficients
Exact equations 31
The linear equation of order one 36

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27

vii


viii

Contents

12. The general solution of a linear equation

Miscellaneous exercises


3

39

42

Elementary Applications
13.
14.
15.
16.
17.

Velocity of escape from the earth 45
Newton's law of cooling 47
Simple chemical conversion 48
Logistic growth and the price of commodities
Orthogonal trajectories 57

53

4 Additional Topics on Equations of Order One
18.
19.
20.
21.
22.
23.

5


Integrating factors found by inspection 61
The determination of integrating factors 65
Substitution suggested by the equation 70
Bernoulli's equation 72
Coefficients linear in the two variables 75
Solutions involving nonelementary integrals 80
Miscellaneous exercises 82

Linear Differential Equations
24. The general linear equation 84
25. Linearindependence 85
26. An existence and uniqueness theorem 86
27. The Wronskian 86
28.
29.
30.
31.
32.

6

General solution of a homogeneous equation

89

General solution of a nonhomogeneous equation 91
Differential operators 92
The fundamental laws of operation 95
Some properties of differential operators 96


Linear Equations with Constant Coefficients
33. Introduction 100
34. The auxiliary equation; distinct roots 100
35. The auxiliary equation; repeated roots 103
36. A definition of exp z for imaginary z 107
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Contents

37. The auxiliary equation: imaginary roots
38. A note on hyperbolic functions 110
Miscellaneous exercises 114

ix

108

7 Nonhomogeneous Equations: Undetermined Coefficients
39. Construction of a homogeneous equation

from a specified solution

116

40. Solution of a nonhomogeneous equation
41. The method of undetermined coefficients
42. Solution by inspection 127


8

Variation of Parameters
43.
44.
45.
46.

9

Introduction 133
Reduction of order 134
Variation of parameters 138
Solution of y" + y = f(x) 142
Miscellaneous exercises 145

Inverse Differential Operators
47. The exponential shift

146

48. The operator l/f(D) 150
49. Evaluation of [1/f(D)]eax 151
50. Evaluation of (D 2 + a 2 )- 1 sin ax and (D 2

10

Applications
51.
52.

53.
54.
55.

11

119
121

Vibration of a spring 156
Undamped vibrations 158
Resonance 161
Damped vibrations 163
The simple pendulum 168

The Laplace Transform
56. The transform concept

170

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+ a 2 )- 1 cos ax 152


x

Contents

Definition of the Laplace transform 171

Transforms of elementary functions 172
Sectionally continuous functions 176
Functions of exponential order 178
Functions of class A 181
Transforms of derivatives 183
63. Derivatives of transforms 186
64. The gamma function 187
65. Periodic functions 188

57.
58.
59.
60.
61.
62.

12

Inverse Transforms
Definition of an inverse transform 194
Partial fractions 198
Initial value problems 201
A step function 206
A convolution theorem 213
Special integral equations 218
Transform methods and the vibration of springs
73. The deflection of beams 226

66.
67.

68.
69.
70.
71.
72.

13

223

Linear Systems of Equations
74. Introduction 233
75. Elementary elimination calculus

233
76. First order systems with constant coefficients
77. Solution of a first order system 239
78. Some matrix algebra 240
79. First-order systems revisted 247
80. Complex eigenvalues 256
81. Repeated eigenvalues 261
82. Nonhomogeneous systems 269
83. Arms races 273
84. The Laplace transform 278

14

Electric Circuits and Networks
85. Circuits 284
86. Simple networks


287

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237


xi

Contents

15

The Existence and Uniqueness of Solutions
87.
88.
89.
90.
91.
92.

16

Preliminary remarks 297
An existence and uniqueness theorem 298
A Lipschitz condition 300
A proof of the existence theorem 301
A proof of the uniqueness theorem 304
Other existence theorems 306


Nonlinear Equations
93.
94.
95.
96.
92.
98.
99.
100.
101.
102.

17

316

Power Series Solutions
103.
104.
105.
106.
107.

18

Preliminary remarks 307
Factoring the left member 308
Singular solutions 311
The c-discriminant equation 312

The p-discriminant equation 314
Eliminating the dependent variable
Clairaut's equation 318
Dependent variable missing 321
Independent variable missing 323
The catenary 326
Miscellaneous exercises 328

Linear equations and power series 330
Convergence of power series 332
Ordinary points and singular points 334
Validity of the solutions near an ordinary point
Solutions near an ordinary point 336

335

Solutions Near Regular Singular Points
108. Regular singular points 347
109. The indicial equation 350
110. Form and validity of the solutions near a regular
singular point 352
111. Indicial equation with difference of roots nonintegral
112. Differentiation of a product of functions 358
113. Indicial equation with equal roots 359

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352



xii

Contents
114. Indicial equation with equal roots, an alternative 365
115. Indicial equation with difference of roots a positive

integer, nonlogarithmic case

369

116. Indicial equation with difference of roots a positive

integer, logarithmic case

374

117. Solution for large x 379
118. Many-term recurrence relations
119. Summary 388

Miscellaneous exercises

19

Equations to be treated in this chapter 392
The factorial function 392
The hypergeometric equation 393
Laguerre polynomials 395
Bessel's equation with index not an integer 396
Bessel's equation with index an integer 397

Hermite polynomials 399
Legendre polynomials 399
The confluent hypergeometric equation 401

Numerical Methods
129.
130.
131.
132.
133.
134.
135.

21

389

Equations of Hypergeometric Type
120.
121.
122.
123.
124.
125.
126.
127.
128.

20


383

General remarks 403
The increment method 404
A method of successive approximation 406
An improvement on the preceding method 408
The use of Taylor's theorem 409
The Runge-Kutta method 412
A continuing method 415

Partial Differential Equations
136.
137.
138.
139.

Remarks on partial differential equations 419
Some partial differential equations of applied mathematics
Method of separation of variables 422
A problem on the conduction of heat in a slab 427
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420


Contents

22

Orthogonal Sets

140.
141.
142.
143.
144.
145.

23

Orthogonality 433
Simple sets of polynomials 434
Orthogonal polynomials 435
Zeros of orthogonal polynomials 436
Orthogonality of Legendre polynomials 437
Other orthogonal sets 439

Fourier Series
146.
147.
148.
149.
150.
151.
152.

24

xiii

Orthogonality of a set of sines and cosines 441

Fourier series: an expansion theorem 444
Numerical examples of Fourier series 448
Fourier sine series 457
Fourier cosine series 461
Numerical Fourier analysis 465
Improvement in rapidity of convergence 466

Boundary Value Problems
153. The one-dimensional heat equation 468
154. Experimental verification of the validity
155.
156.
157.
158.

25

Additional Properties of the Laplace Transform
159.
160.
161.
162.

26

of the heat equation 474
Surface temperature varying with time 477
Heat conduction in a sphere 479
The simple wave equation 481
Laplace's equation in two dimensions 484


Power series and inverse transforms 488
The error function 492
Bessel functions 499
Differential equations with variable coefficients

502

Partial Differential Equations; Transform Methods
163. Boundary value problems

503

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xiv

Contents
164.
165.
166.
167.
168.

The wave equation 508
Diffusion in a semi-infinite solid 510
Canonical variables 513
Diffusion in a slab of finite width 515
Diffusion in a quarter-infinite solid 519


Index

523

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1
Definitions, Elimination of
Arbitrary Constants

1. Examples of differential equations
In physics, engineering, and chemistry and, increasingly, in such subjects
as biology, physiology, and economics it is necessary to build a mathematical
model to represent certain problems. It is often the case that these mathematical models involve the search for an unknown function that satisfies an
equation in which derivatives of the unknown function play an important
role. Such equations are called differential equations. As in equation (3)
below, a derivative may be involved implicitly through the presence of
differentials. Our aim is to find methods for solving differential equations;
that is, to find the unknown function or functions that satisfy the differential
equation.
The following are examples of differential equations:
dy
(1)
dx =cos x,
d2y
dx2

+k


2

-

y - 0,

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(2)

1


2

Definitions, Eliminations of Arbitrary Constants
(x 2

+ y 2)dx

- 2xydy

= 0,

au= h2(a 2u + a 1 u)
at
ax 2 ay 2 '
d 2i


1

di

Ldl
+ Rdt + -i
t
c =
a1 v
ax2

+

(ddx2wr 2

d 3x
dy 3
d2y
dx2

a1 v
ay2

dw
xy dx
dx

( dy)
7 dx


d 2y
dt 2
af
x ax

+

3

+w=

d 2x
dt2

0,

= 0,

- 8y

= 0,

(4)

(5)

(7)

(8)


(9)
(10)

= x,

af

+ y ay =

(3)

(6)

= 0,

~ 4xy

+ x dy
+

Ew cos wt,

(Ch.1

(11)

nf.

When an equation involves one or more derivatives with respect to a
particular variable, that variable is called an independent variable. A variable

is called dependent if a derivative of that variable occurs.
In the equation
(5)

i is the dependent variable, t the independent variable, and L, R, C, E, and
w are called parameters. The equation

a1 v
ax2

+

a1 v

(6)

ay2 = 0

has one dependent variable V and two independent variables.
Since the equation
(x 2

+ y 2 ) dx

- 2xy dy

=0

may be written
x2


+ y2

-

dy
2xy- = 0
dx

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(3)


§ 2)

3

Definitions

or

dx

+ y 2 ) dy - 2xy =

(x 2

0,


we may consider either variable to be dependent, the other being the independent one.

Exercise
Identify the independent variables, the dependent variables, and the parameters in
the equations given as examples in this section.

2. Definitions
The order of a differential equation is the order of the highest-ordered
derivative appearing in the equation. For instance,

d2y
dx2

3
+ 2b (dy)
dx + y =

(1)

0

is an equation of "order two." It is also referred to as a "second-order
equation."
More generally, the equation
F(x, y, y', .. . , y
=0

(2)

is called an "nth-order" ordinary differential equation. Under suitable
restrictions on the function F, equation (2) can be solved explicitly for y<n> in
terms of the other n + 1 variables x, y, y', ... , y<n- 1 >, to obtain

y
(3)

For the purposes of this book we shall assume that this is always possible.
Otherwise, an equation of the form of equation (2) may actually represent
more than one equation of the form of equation (3).
For example, the equation
x(y') 2

+ 4y'

- 6x 2

=0

actually represents the two different equations,
y

,

=

-2

+ J4 + 6x 3

x

or

y'

14 + 6x 3
-2 - '\/~--= ___

x

A function ¢, defined on an interval a < x < b, is called a solution of the
differential equation (3), provided that the n derivatives of the function exist

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4

(Ch.1

Definitions, Eliminations of Arbitrary Constants

on the interval a < x < b and
cp<">(x) = f(x, cp(x), cp'(x), ... , Â<ã- 1 >(x)),

for every x in a < x < b.
For example, let us verify that
y

=

e1x

is a solution of the equation
d1y
dx 1

dy

+ dx

- 6Y

(4)

= O.

We substitute our tentative solution into the left member of equation (4)
and find that
d1 y

-

dx 1

dy

+- dx

6y

= 4 e 1x + 2 e 1 x

-

6 e 1x

=0,

which completes the desired verification.
All of the equations we shall consider in Chapter 2 are of order one, and
hence may be written
dy
dx

=

f(x,y).

For such equations it is sometimes convenient to use the definitions of elementary calculus to write the equation in the form
M(x, y) dx

+ N(x, y) dy =

0.

(5)

A very important concept in the study of differential equations is that of

linearity. An ordinary differential equation of order n is called linear if it
may be written in the form
d"y
bo(x)-d
x"

d"- 1 y

dy

+ b 1 (x)d----=-1
+ · · · + bn- l (x)-dx + bn(x)y =
x"

R(x).

For example, equation (1) above is nonlinear, and equation (4) is linear.
The equation
x 1 y"

+ xy' + (x 2

-

n 1 )y

= 4x 3

is also linear.
The notion of linearity may be extended to partial differential equations.

For example, the equation

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§ 3]

5

The elimination of arbitrary constants

is the general first-order linear partial differential equation with two independent variables and
o2w
bo(X, y) OXZ

o2w

o2w

+ b1(X, y) OX oy + hz(X, y) oy2
ow

ow

+ b3(X, y) OX + b4(X, y) oy + bs(X, y)w =

R(x, y)

is the general second-order linear partial differential equation with two
independent variables.

Exercises
For each of the following, state whether the equation is ordinary or partial, linear
or nonlinear, and give its order.
d 2x

+ k2x =

1. dt2

3. (x 2
5. y"'

0.

+ y 2)dx + 2xydy =
- 3y' + 2y = 0.

o2u
7. ax2

o 2u

d2 y
d2 x
9. X dt2 - y dt2 =
11. (x

13.

d4y
8. dx4

0.

di

10. Ldt

C1.

+ y)dx + (3x 2

_

=

-

l)dy = 0.

(~:~r 2(~:r + yw = o.

15. y"

+ 2y' -

Sy= x 2

+ cosx.

Q(x).

6. yy" = x.

o2u

+ 8y2 + oz2

+ P(x)y =

4. y'

0.

=

w(x).

.

+ Ri =

12. x(y") 3

E.

+ (y') 4

dy

14. dx = 1 - xy
16. a da

+ b db

=

-

y = 0.

+y

2

.

0.

3. The elimination of arbitrary constants
In practice, differential equations arise in many ways, some of which we
shall encounter later. There is one way of arriving at a differential equation,
however, that is useful in that it gives us a feeling for the kinds of solutions to
be expected. In this section we shall start with a relation involving arbitrary
constants and, by elimination of those arbitrary constants, come to a differential equation consistent with the original relation. In a sense we start with
the answer and find the problem.
Methods for the elimination of arbitrary constants vary with the way in
which the constants enter the given relation. A method that is efficient for
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6

Definitions, Eliminations of Arbitrary Constants

[Ch. 1

one problem may be poor for another. One fact persists throughout. Because
each differentiation yields a new relation, the number of derivatives that need
be used is the same as the number of arbitrary constants to be eliminated.
We shall in each case determine the differential equation that is
(a) Of order equal to the number of arbitrary constants in the given
relation.
(b) Consistent with that relation.
(c) Free from arbitrary constants.
EXAMPLE (a):
the relation

Eliminate* the arbitrary constants c 1 and c2 from

+ C2 e3x.

y = C1 e-2x

(1)

Since two constants are to be eliminated, obtain the two derivatives,

y' = -2c 1 e-ix

y" = 4c1

e-2x

+ 3c 2 e 3 X,

+ 9c2 e3x.

(2)

(3)

The elimination of c 1 from equations (2) and (3) yields
y"

+ 2y' =

15c 2 e3 x;

the elimination of c 1 from equations (1) and (2) yields
y'

+ 2y

= 5c 2 e 3 x.

Hence
y"

+ 2y' =

3(y'

+ 2y),

or
y" - y' - 6y

= 0.

Another method for obtaining the differential equation in this example
proceeds as follows : We know from a theorem in elementary algebra that
the three equations (1), (2), and (3), considered as equations in the two unknowns c 1 and c2 , can have solutions only if
e-2x

-y
-y'
-y"

-

e3x

2 e -2x

3 e3x = 0.

4 e-2x

9 e3x

(4)

Since e- 2 x and e 3 x cannot be zero, equation (4) may be rewritten, with the
factors - 1, e - 2 X, and e3 x removed, as

* By differentiations and pertinent legitimate mathematical procedures. Elimination by
erasure, for instance, is not permitted.
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§ 3)

The elimination of arbitrary constants

y

1

y'

-2

y"

4

7

1

3 = 0
9

from which the differential equation

y" - y' - 6y = 0
follows immediately.
This latter method has the advantage of making it easy to see that the
elimination of the constants c 1 , c2 , .•• , en from a relation of the form

will always lead to a linear differential equation

dny
aodxn

dn-ly

dy

+ aldxn-1 + ... + an-ldx +any= 0,

in which the coefficients a0 , a 1 , . . . , an are constants. The study of such
differential equations will receive much of our attention.

EXAMPLE (b):

Eliminate the constant a from the equation

+ y2 =

(x - a)2

a1.

Direct differentiation of the relation yields
2(x - a)

+ 2yy' =

0,

from which

a= x

+ yy'.

Therefore, using the original equation, we find that

(yy')2

+ y2 =

(x

+ yy')2,

.or

y2 = x 2

+ 2xyy',

which may be written in the form
(x 2

-

y 2)dx

+ 2xydy = 0.

Another method will be used in this example as an illustration of a device
that is often helpful. The method is based upon the isolation of an arbitrary
constant.

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8

Definitions, Eliminations of Arbitrary Constants

(Ch.1

The equation

+ y2 =

(x - a)2

az

may be put in the form

+ y2

x2

2ax

-

= 0,

or
x2

+ y2

---=

x

2a.

Then differentiation of both members leads to
x(2x dx

+ 2y dy)

- (x 2

+ y 2 ) dx =

x2

0

,

or
(x 2

y 2 )dx

-

+ 2xydy =

0,

as desired.
It is interesting to speculate here about the significance of x = 0 upon the
argument just used. The student should draw a few of the members of this
family of circles and observe what is peculiar about their behavior at x = 0.
EXAMPLE (c):

Eliminate Band IX from the relation

x

= B cos (wt + IX),

(5)

in which w is a parameter (not to be eliminated).
First we obtain two derivatives of x with respect to t:
dx
dt
d2 x
dt 2

-wB sin (wt+ IX),

(6)

-w 2 B cos (wt + IX).

(7)

Comparison of equations (5) and (7) shows at once that

EXAMPLE (d):

Eliminate c from the equation

+ c2 x + 4 =

0.

+ xy') + c 2 =

0.

cxy
At once we get
c(y

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§ 3)

The eliminations of arbitrary constants

9

Since c =I- 0,
c = -(y

+ xy')

and substitution into the original equation leads us to the result
x3(y')2

+ x2yy' + 4 =

0.

Our examples suggest that in a certain sense the totality of solutions of an
nth-order equation depends on n arbitrary constants. The sense in which this
is true will be stated in Sections 5 and 12.

Exercises
In each of the following eliminate the arbitrary constants.

1.
2.
3.
4;

xsiny + x 2 y = c.
3x 2 - xy 2 = c.
xy 2 - 1 = cy.
cx 2 + x + y 2 = 0.

ANS.

(sin y

+ 2xy) dx + (x cosy + x 2) dy
ANS.
ANS.
ANS.

= 0.
(6x - y 2 ) dx - 2xy dy = 0.
y 3 dx + (xy 2 + 1) dy = 0.
(x + 2y 2 ) dx - 2xy dy = 0.

5. x =A sin (wt+ /3); w a parameter, not to be eliminated.
6. x = c 1 cos wt
7. y =ex

8. y
9.
10.
11.
12.
13.
14.
15.
16.
17.
18.
19.
'":. 20.
21.
22.
23.
24.

25.

+ c 2 sin wt; w a parameter.

+ c2 + 1.

= .mx +!!_;ha

m

ANS.

+w

2

-

x - 0.

ANS.

ANS.

parameter, m to be eliminated.

dzx
dt2

y = xy'
ANS.

+ (y') 2 + 1.

y = xy '

h

+ ---;.

y

y 2 = 4ax.
ANS. 2xdy - ydx = 0.
y = ax 2 + bx + c.
ANS. y"' = 0.
y = C1 + Cz e 3 x.
ANS. y" - 3y' = 0.
y = 4 + c 1 e 3 x.
ANS. y' - 3y = -12.
y = c 1 + c 2 e- 4 x.
ANS. y" + 4y' = 0.
y = C1 ex + Cz e-x.
ANS. y" - y = 0.
y = X + C1 ex+ C2 e-x.
ANS. y" - y = -x.
y = C1 e 2 x + Cz e 3 x.
ANS. y" - Sy' + 6y = 0.
y = x 2 + c 1 e 2 x + c 2 e 3 x.
ANS. y" - Sy' + 6y = 6x 2 - lOx + 2.
y = c 1 ex+ c 2 xex.
ANS. y" - 2y' + y = 0.
y =A e 2 x + Bx e 2 x.
ANS. y" - 4y' + 4y = 0.
y = c 1 e 2 x cos 3x + c 2 e 2 x sin 3x.
ANS. y" - 4y' + 13y = 0.
y = c 1 e•x cos bx + c 2 e•x sin bx; a and bare parameters.
ANS. y" - 2ay' + (a 2 + b 2 )y = 0.

y = c 1 x + c 2 ex.
ANS. (x - l)y" - xy' + y = 0.
y = c 1 x 2 + c 2 e-x.
ANS. x(x + 2)y" + (x 2 - 2)y' - 2(x + l)y = 0.
y = x 2 + c 1 x + c 2 e-x.
ANS. (x + l)y" + xy' - y = x 2 + 2x + 2.
Y = c 1 x 2 + c 2 e 2 x.
ANS. x(l - x)y" + (2x 2 - l)y' - 2(2x - l)y = 0.

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10

[Ch.1

Definitions, Eliminations of Arbitrary Constants

4. Families of curves
An equation involving a parameter, as well as one or both of the coordinates of a point in a plane, may represent a family of curves, one curve
corresponding to each value of the parameter. For instance, the equation
(x - c) 2

+ (y

- c) 2

=

(1)

2c 2 ,

or
x2

+ y2

-

2c(x

+ y) =

(2)

0,

may be interpreted as the equation of a family of circles, each having its
center on the line y = x and each passing through the origin. Figure 1 shows
several elements, or members, of this family.
y

FIGURE 1

If the constant c in equation (1) or in equation (2) is treated as an arbitrary
constant and eliminated as in the preceding section, the result is called the
differential equation of the family represented by equation (1). In this example,
the elimination of c is easily performed by isolating c, then differentiating
throughout the equation with respect to x. Thus, from

x2 + y2
x+y

- - - = 2c

we find that

_(x_+_y_)(_2x_d_x_+_2_y_dy_)_-_(x_2 _+_y_2_)(_dx_+_d_y) = 0
(x + y)2
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.


§ 4)

11

Families of curves

Therefore
(x 2

+ 2xy

- y 2 ) dx - (x 2

-

2xy - y 2 ) dy

=

0

(3)

is the differential equation of the family of circles represented by equation (1).
Note that equation (3) associates a definite slope with each point (x, y) in
the plane
dy
dx

x2
x2

=

+ 2xy
-

- y2
2xy - y 2

(4)

'

except where the denominator on the right in equation (4) vanishes. When

the denominator vanishes, the curve passing through that point must have
a vertical tangent. From
x2

-

2xy - y 2

=0

we see that

y = (-1

+ j2)x

(5)

or

y = ( - 1 - j2)x.

(6)

In Figure 2, the straight lines (5) and (6) appear along with the family (1).
It is seen that the lines (5) and (6) cut the members of the family of circles in
precisely those points of vertical tangency.

y
y=x

y=(-1-J2)x

y=(-1+v'2)x

FIGURE 2

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