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Differential
Equations
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Differential
Equations
Fourth Edition
Richard Bronson, PhD
Professor of Mathematics and Computer Science
Fairleigh Dickinson University
Gabriel B. Costa, PhD
Professor of Mathematical Sciences
United States Military Academy
Associate Professor of Mathematics and Computer Science
Seton Hall University
Schaum’s Outline Series
New York Chicago San Francisco Athens London Madrid
Mexico City Milan New Delhi Singapore Sydney Toronto
RICHARD BRONSON, PhD is Professor of Mathematics at Fairleigh Dickinson University. He received his PhD in applied mathematics
from the Stevens Institute of Technology in 1968. Dr Bronson has served as an associate editor of the journal Simulation, as a contributing
editor to SIAM News, and as a consultant to Bell Laboratories. He has conducted joint research in mathematical modeling and computer
simulation at the Technion–Israel Institute of Technology and the Wharton School of Business at the University of Pennsylvania. Dr Bronson
has published over 30 technical articles and books, the latter including x Schaum’s Outline of Matrix Operations and Schaum’s Outline of
Operations Research.
GABRIEL B. COSTA, PhD is a Catholic priest and Professor of Mathematical Sciences at the United States d Military Academy, West
Point, NY, where he also functions as an associate chaplain. Father Costa is on extended leave from Seton Hall University, South Orange,
NJ. He received his PhD in the area of differential equations from Stevens Institute of Technology in 1984. In addition to differential
equations, Father Costa’s academic interests include mathematics education and sabermetrics, the search for objective knowledge about
baseball.
Copyright © 2014 by McGraw-Hill Education. All rights reserved. Except as permitted under the United States Copyright Act of 1976, no
part of this publication may be reproduced or distributed in any form or by any means, or stored in a database or retrieval system, without
the prior written permission of the publisher.
ISBN: 978-0-07-182286-2
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whether such claim or cause arises in contract, tort or otherwise.
To Ignace and Gwendolyn Bronson, Samuel and Rose Feldschuh — RB
To the great mathematicians and educators I have been blessed to meet: Professors Bloom, Brady, Bronson,
Dostal, Goldfarb, Levine, Manogue, Pinkham, Pollara and Suffel…and, of course, Mr Rod! — GBC
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vii
PREFACE
Differential equations are among the linchpins of modern mathematics which, along with matrices, are
essential for analyzing and solving complex problems in engineering, the natural sciences, econom-
ics, and even business. The emergence of low-cost, high-speed computers has spawned new tech-
niques for solving differential equations, which allows problem solvers to model and solve complex
problems based on systems of differential equations.
As with the two previous editions, this book outlines both the classical theory of differential equa-
tions and a myriad of solution techniques, including matrices, series methods, Laplace transforms and
several numerical methods. We have added a chapter on modeling and touch upon some qualitative
methods that can be used when analytical solutions are difficult to obtain. A chapter on classical dif-
ferential equations (e.g., the equations of Hermite, Legendre, etc.) has been added to give the reader
exposure to this rich, historical area of mathematics.
This edition also features a chapter on difference equations and parallels this with differential
equations. Furthermore, we give the reader an introduction to partial differential equations and the
solution techniques of basic integration and separation of variables. Finally, we include an appendix
dealing with technology touching upon the TI-89 hand-held calculator and the MATHEMATICA
software packages.
With regard to both solved and supplementary problems, we have added such topics as integral
equations of convolution type, Fibonacci numbers, harmonic functions, the heat equation and the wave
equation. We have also alluded to both orthogonality and weight functions with respect to classical
differential equations and their polynomial solutions. We have retained the emphasis on both initial
value problems and differential equations without subsidiary conditions. It is our aim to touch upon
virtually every type of problem the student might encounter in a one-semester course on differential
equations.
Each chapter of the book is divided into three parts. The first outlines salient points of the theory
and concisely summarizes solution procedures, drawing attention to potential difficulties and sub-
tleties that too easily can be overlooked. The second part consists of worked-out problems to clarify and,
on occasion, to augment the material presented in the first part. Finally, there is a section of problems
with answers that readers can use to test their understanding of the material.
The authors would like to thank the following individuals for their support and invaluable assis-
tance regarding this book. We could not have moved as expeditiously as we did without their support
and encouragement. We are particularly indebted to Dean John Snyder and Dr. Alfredo Tan of
Fairleigh Dickinson University. The continued support of the Most Reverend John J Myers, J.C.D.,
D.D., Archbishop of Newark, N.J., is also acknowledged. From Seton Hall University we are grateful
to the Reverend Monsignor James M. Cafone and to the members of the Priest Community; we also
thank Dr. Fredrick Travis, Dr. James Van Oosting, Dr. Molly Smith, and Dr. Bert Wachsmuth and the
members of the Department of Mathematics and Computer Science. We also thank Colonel Gary
W. Krahn of the United States Military Academy.
Ms. Barbara Gilson and Ms. Adrinda Kelly of McGraw-Hill were always ready to provide any
needed guidance and Dr. Carol Cooper, our contact in the United Kingdom, was equally helpful.
Thank you, one and all.
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Chapter 1
Basic Concepts
1
Differential Equations
1
Notation
2
Solutions
2
Initial-Value and Boundary-Value Problems
2
Chapter 2
An Introduction to Modeling and
Qualitative Methods
9
Mathematical Models
9
The “Modeling Cycle”
9
Qualitative Methods
10
Chapter 3
Classifications of First-Order
Differential Equations
14
Standard Form and Differential Form
14
Linear Equations
14
Bernoulli Equations
14
Homogeneous Equations
15
Separable Equations
15
Exact Equations
15
Chapter 4
Separable First-Order Differential Equations
21
General Solution
21
Solutions to the Initial-Value Problem
21
Reduction of Homogeneous Equations
22
Chapter 5
Exact First-Order Differential Equations
31
Defining Properties
31
Method of Solution
31
Integrating Factors
32
Chapter 6
Linear First-Order Differential Equations
42
Method of Solution
42
Reduction of Bernoulli Equations
42
ix
CONTENTS
Chapter 7
Applications of First-Order Differential
Equations
50
Growth and Decay Problems
50
Temperature Problems
50
Falling Body Problems
51
Dilution Problems
52
Electrical Circuits
52
Orthogonal Trajectories
53
Chapter 8
Linear Differential Equations: Theory of
Solutions
73
Linear Differential Equations
73
Linearly Independent Solutions
74
The Wronskian
74
Nonhomogeneous Equations
74
Chapter 9
Second-Order Linear Homogeneous Differential
Equations with Constant Coefficients
83
Introductory Remark
83
The Characteristic Equation
83
The General Solution
84
Chapter 10
nth-Order Linear Homogeneous Differential
Equations with Constant Coefficients
89
The Characteristic Equation
89
The General Solution
90
Chapter 11
The Method of Undetermined Coefficients
94
Simple Form of the Method
94
Generalizations
95
Modifications
95
Limitations of the Method
95
Chapter 12
Variation of Parameters
103
The Method
103
Scope of the Method
104
Chapter 13
Initial-Value Problems for Linear
Differential Equations
110
x
CONTENTS
Chapter 14
Applications of Second-Order Linear
Differential Equations
114
Spring Problems
114
Electrical Circuit Problems
115
Buoyancy Problems
116
Classifying Solutions
117
Chapter 15
Matrices
131
Matrices and Vectors
131
Matrix Addition
131
Scalar and Matrix Multiplication
132
Powers of a Square Matrix
132
Differentiation and Integration of Matrices
132
The Characteristic Equation
133
Chapter 16
eAt
140
Definition
140
Computation of eAt
140
Chapter 17
Reduction of Linear Differential Equations
to a System of First-Order Equations
148
An Example
148
Reduction of an nth Order Equation
149
Reduction of a System
150
Chapter 18
Graphical and Numerical Methods for Solving
First-Order Differential Equations
157
Qualitative Methods
157
Direction Fields
157
Euler’s Method
158
Stability
158
Chapter 19
Further Numerical Methods for Solving
First-Order Differential Equations
176
General Remarks
176
Modified Euler’s Method
177
Runge–Kutta Method
177
Adams–Bashford–Moulton Method
177
Milne’s Method
177
Starting Values
178
Order of a Numerical Method
178
CONTENTS
xi
Chapter 20
Numerical Methods for Solving Second-Order
Differential Equations Via Systems
195
Second-Order Differential Equations
195
Euler’s Method
196
Runge–Kutta Method
196
Adams–Bashford–Moulton Method
196
Chapter 21
The Laplace Transform
211
Definition
211
Properties of Laplace Transforms
211
Functions of Other Independent Variables
212
Chapter 22
Inverse Laplace Transforms
224
Definition
224
Manipulating Denominators
224
Manipulating Numerators
225
Chapter 23
Convolutions and the Unit Step Function
233
Convolutions
233
Unit Step Function
233
Translations
234
Chapter 24
Solutions of Linear Differential Equations with
Constant Coefficients by Laplace Transforms
242
Laplace Transforms of Derivatives
242
Solutions of Differential Equations
243
Chapter 25
Solutions of Linear Systems by Laplace
Transforms
249
The Method
249
Chapter 26
Solutions of Linear Differential Equations
with Constant Coefficients by Matrix Methods
254
Solution of the Initial-Value Problem
254
Solution with No Initial Conditions
255
Chapter 27
Power Series Solutions of Linear Differential
Equations with Variable Coefficients
262
Second-Order Equations
262
Analytic Functions and Ordinary Points
262
Solutions Around the Origin of Homogeneous Equations
263
xii
CONTENTS
Solutions Around the Origin of Nonhomogeneous Equations
263
Initial-Value Problems
264
Solutions Around Other Points
264
Chapter 28
Series Solutions Near a Regular Singular Point
275
Regular Singular Points
275
Method of Frobenius
275
General Solution
276
Chapter 29
Some Classical Differential Equations
290
Classical Differential Equations
290
Polynomial Solutions and Associated Concepts
290
Chapter 30
Gamma and Bessel Functions
295
Gamma Function
295
Bessel Functions
295
Algebraic Operations on Infinite Series
296
Chapter 31
An Introduction to Partial
Differential Equations
304
Introductory Concepts
304
Solutions and Solution Techniques
305
Chapter 32
Second-Order Boundary-Value Problems
309
Standard Form
309
Solutions
310
Eigenvalue Problems
310
Sturm–Liouville Problems
310
Properties of Sturm–Liouville Problems
310
Chapter 33
Eigenfunction Expansions
318
Piecewise Smooth Functions
318
Fourier Sine Series
319
Fourier Cosine Series
319
Chapter 34
An Introduction to Difference Equations
325
Introduction
325
Classifications
325
Solutions
326
APPENDIX A
Laplace Transforms
330
CONTENTS
xiii
