Elementary Differential Equations

1981·8.14 MB·English

by  Earl D. Rainville

Elementary Differential Equations
Elementary
Dijfe rential
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
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
Edward, joint author. II. Title.
QA371.R29 1980 515.3'5 80-12849
ISBN 0-02-397770-1
Printing: 12345678 Year: 12345678
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 neces
sary 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.
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
Contents
1 Definitions, Elimination of Arbitrary Constants
1. Examples of differential equations
2. Definitions 3
3. The elimination of arbitrary constants 5
4. Families of curves 10
2 Equations of Order One
5. The isoclines of an equation 16
6. An existence theorem 19
7. Separation of variables 20
8. Homogeneous functions 25
9. Equations with homogeneous coefficients 27
10. Exact equations 31
11. The linear equation of order one 36
vii
viii Contents
12. The general solution of a linear equation 39
Miscellaneous exercises 42
3 Elementary Applications
13. Velocity of escape from the earth 45
14. Newton's law of cooling 47
15. Simple chemical conversion 48
16. Logistic growth and the price of commodities 53
17. Orthogonal trajectories 57
4 Additional Topics on Equations of Order One
18. Integrating factors found by inspection 61
19. The determination of integrating factors 65
20. Substitution suggested by the equation 70
21. Bernoulli's equation 72
22. Coefficients linear in the two variables 75
23. Solutions involving nonelementary integrals 80
Miscellaneous exercises 82
5 Linear Differential Equations
24. The general linear equation 84
25. Linearindependence 85
26. An existence and uniqueness theorem 86
27. The Wronskian 86
28. General solution of a homogeneous equation 89
29. General solution of a nonhomogeneous equation 91
30. Differential operators 92
31. The fundamental laws of operation 95
32. Some properties of differential operators 96
6 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
Contents ix
37. The auxiliary equation: imaginary roots 108
38. A note on hyperbolic functions 110
Miscellaneous exercises 114
7 Nonhomogeneous Equations: Undetermined Coefficients
39. Construction of a homogeneous equation
from a specified solution 116
40. Solution of a nonhomogeneous equation 119
41. The method of undetermined coefficients 121
42. Solution by inspection 127
8 Variation of Parameters
43. Introduction 133
44. Reduction of order 134
45. Variation of parameters 138
46. Solution of y" + y = f(x) 142
Miscellaneous exercises 145
9 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 (D2 + a2)-1 sin ax and (D2 + a2)-1 cos ax 152
10 Applications
51. Vibration of a spring 156
52. Undamped vibrations 158
53. Resonance 161
54. Damped vibrations 163
55. The simple pendulum 168
11 The Laplace Transform
56. The transform concept 170
x Contents
57. Definition of the Laplace transform 171
58. Transforms of elementary functions 172
59. Sectionally continuous functions 176
60. Functions of exponential order 178
61. Functions of class A 181
62. Transforms of derivatives 183
63. Derivatives of transforms 186
64. The gamma function 187
65. Periodic functions 188
12 Inverse Transforms
66. Definition of an inverse transform 194
67. Partial fractions 198
68. Initial value problems 201
69. A step function 206
70. A convolution theorem 213
71. Special integral equations 218
72. Transform methods and the vibration of springs 223
73. The deflection of beams 226
13 Linear Systems of Equations
74. Introduction 233
75. Elementary elimination calculus 233
76. First order systems with constant coefficients 237
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