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Differential equations with boundary-value problems
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REVIEW OF DIFFERENTIATION
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BRIEF TABLE OF INTEGRALS
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Eighth Edition
DIFFERENTIAL
EQUATIONS
with Boundary-Value Problems
DENNIS G. ZILL
Loyola Marymount University
WARREN S. WRIGHT
Loyola Marymount University
MICHAEL R. CULLEN
Late of Loyola Marymount University
Australia • Brazil • Japan • Korea • Mexico • Singapore • Spain • United Kingdom • United States
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Differential Equations with
Boundary-Value Problems,
Eighth Edition
Dennis G. Zill, Warren S. Wright,
and Michael R. Cullen
Publisher: Richard Stratton
Senior Sponsoring Editor:
Molly Taylor
Development Editor: Leslie Lahr
Assistant Editor:
Shaylin Walsh Hogan
Editorial Assistant: Alex Gontar
Media Editor: Andrew Coppola
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Cover Image: ©Wally Pacholka
Compositor: MPS Limited,
a Macmillan Company
Section 4.8 of this text appears in
Advanced Engineering Mathematics,
Fourth Edition, Copyright 2011,
Jones & Bartlett Learning, Burlington,
MA 01803 and is used with the
permission of the publisher.
© 2013, 2009, 2005 Brooks/Cole, Cengage Learning
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Library of Congress Control Number: 2011944305
ISBN-13: 978-1-111-82706-9
ISBN-10: 1-111-82706-0
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3
v
Contents
1 INTRODUCTION TO DIFFERENTIAL EQUATIONS 1
Preface xi
Projects P-1
1.1 Definitions and Terminology 2
1.2 Initial-Value Problems 13
1.3 Differential Equations as Mathematical Models 20
Chapter 1 in Review 33
2 FIRST-ORDER DIFFERENTIAL EQUATIONS 35
2.1 Solution Curves Without a Solution 36
2.1.1 Direction Fields 36
2.1.2 Autonomous First-Order DEs 38
2.2 Separable Equations 46
2.3 Linear Equations 54
2.4 Exact Equations 63
2.5 Solutions by Substitutions 71
2.6 A Numerical Method 75
Chapter 2 in Review 80
MODELING WITH FIRST-ORDER DIFFERENTIAL EQUATIONS 83
3.1 Linear Models 84
3.2 Nonlinear Models 95
3.3 Modeling with Systems of First-Order DEs 106
Chapter 3 in Review 113
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5
4
vi ● CONTENTS
HIGHER-ORDER DIFFERENTIAL EQUATIONS 116
4.1 Preliminary Theory—Linear Equations 117
4.1.1 Initial-Value and Boundary-Value Problems 117
4.1.2 Homogeneous Equations 119
4.1.3 Nonhomogeneous Equations 124
4.2 Reduction of Order 129
4.3 Homogeneous Linear Equations with Constant Coefficient 132
4.4 Undetermined Coefficients—Superposition Approach 139
4.5 Undetermined Coefficients—Annihilator Approach 149
4.6 Variation of Parameters 156
4.7 Cauchy-Euler Equation 162
4.8 Green’s Functions 169
4.8.1 Initial-Value Problems 169
4.8.2 Boundary-Value Problems 176
4.9 Solving Systems of Linear DEs by Elimination 180
4.10 Nonlinear Differential Equations 185
Chapter 4 in Review 190
MODELING WITH HIGHER-ORDER DIFFERENTIAL EQUATIONS 192
5.1 Linear Models: Initial-Value Problems 193
5.1.1 Spring/Mass Systems: Free Undamped Motion 193
5.1.2 Spring/Mass Systems: Free Damped Motion 197
5.1.3 Spring/Mass Systems: Driven Motion 200
5.1.4 Series Circuit Analogue 203
5.2 Linear Models: Boundary-Value Problems 210
5.3 Nonlinear Models 218
Chapter 5 in Review 228
SERIES SOLUTIONS OF LINEAR EQUATIONS 231
6.1 Review of Power Series 232
6.2 Solutions About Ordinary Points 238
6.3 Solutions About Singular Points 247
6.4 Special Functions 257
Chapter 6 in Review 271
6
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CONTENTS ● vii
7 THE LAPLACE TRANSFORM 273
7.1 Definition of the Laplace Transform 274
7.2 Inverse Transforms and Transforms of Derivatives 281
7.2.1 Inverse Transforms 281
7.2.2 Transforms of Derivatives 284
7.3 Operational Properties I 289
7.3.1 Translation on the s-Axis 290
7.3.2 Translation on the t-Axis 293
7.4 Operational Properties II 301
7.4.1 Derivatives of a Transform 301
7.4.2 Transforms of Integrals 302
7.4.3 Transform of a Periodic Function 307
7.5 The Dirac Delta Function 312
7.6 Systems of Linear Differential Equations 315
Chapter 7 in Review 320
8 SYSTEMS OF LINEAR FIRST-ORDER DIFFERENTIAL EQUATIONS 325
8.1 Preliminary Theory—Linear Systems 326
8.2 Homogeneous Linear Systems 333
8.2.1 Distinct Real Eigenvalues 334
8.2.2 Repeated Eigenvalues 337
8.2.3 Complex Eigenvalues 342
8.3 Nonhomogeneous Linear Systems 348
8.3.1 Undetermined Coefficient 348
8.3.2 Variation of Parameters 351
8.4 Matrix Exponential 356
Chapter 8 in Review 360
9 NUMERICAL SOLUTIONS OF ORDINARY DIFFERENTIAL EQUATIONS 362
9.1 Euler Methods and Error Analysis 363
9.2 Runge-Kutta Methods 368
9.3 Multistep Methods 373
9.4 Higher-Order Equations and Systems 375
9.5 Second-Order Boundary-Value Problems 380
Chapter 9 in Review 384
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viii ● CONTENTS
10 PLANE AUTONOMOUS SYSTEMS 385
10.1 Autonomous Systems 386
10.2 Stability of Linear Systems 392
10.3 Linearization and Local Stability 400
10.4 Autonomous Systems as Mathematical Models 410
Chapter 10 in Review 417
11 FOURIER SERIES 419
11.1 Orthogonal Functions 420
11.2 Fourier Series 426
11.3 Fourier Cosine and Sine Series 431
11.4 Sturm-Liouville Problem 439
11.5 Bessel and Legendre Series 446
11.5.1 Fourier-Bessel Series 447
11.5.2 Fourier-Legendre Series 450
Chapter 11 in Review 453
12 BOUNDARY-VALUE PROBLEMS IN RECTANGULAR COORDINATES 455
12.1 Separable Partial Differential Equations 456
12.2 Classical PDEs and Boundary-Value Problems 460
12.3 Heat Equation 466
12.4 Wave Equation 468
12.5 Laplace’s Equation 473
12.6 Nonhomogeneous Boundary-Value Problems 478
12.7 Orthogonal Series Expansions 483
12.8 Higher-Dimensional Problems 488
Chapter 12 in Review 491
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CONTENTS ● ix
13 BOUNDARY-VALUE PROBLEMS IN OTHER COORDINATE SYSTEMS 493
13.1 Polar Coordinates 494
13.2 Polar and Cylindrical Coordinates 499
13.3 Spherical Coordinates 505
Chapter 13 in Review 508
14 INTEGRAL TRANSFORMS 510
14.1 Error Function 511
14.2 Laplace Transform 512
14.3 Fourier Integral 520
14.4 Fourier Transforms 526
Chapter 14 in Review 532
15 NUMERICAL SOLUTIONS OF PARTIAL DIFFERENTIAL EQUATIONS 534
15.1 Laplace’s Equation 535
15.2 Heat Equation 540
15.3 Wave Equation 545
Chapter 15 in Review 549
APPENDIXES
I Gamma Function APP-1
II Matrices APP-3
III Laplace Transforms APP-21
Answers for Selected Odd-Numbered Problems ANS-1
Index I-1
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xi
TO THE STUDENT
Authors of books live with the hope that someone actually reads them. Contrary to
what you might believe, almost everything in a typical college-level mathematics
text is written for you, and not the instructor. True, the topics covered in the text are
chosen to appeal to instructors because they make the decision on whether to use it
in their classes, but everything written in it is aimed directly at you, the student. So
we want to encourage you—no, actually we want to tell you—to read this textbook!
But do not read this text like you would a novel; you should not read it fast and you
should not skip anything. Think of it as a workbook. By this we mean that mathematics should always be read with pencil and paper at the ready because, most likely, you
will have to work your way through the examples and the discussion. Before attempting any of the exercises, work all the examples in a section; the examples are constructed to illustrate what we consider the most important aspects of the section, and
therefore, reflect the procedures necessary to work most of the problems in the exercise sets. We tell our students when reading an example, copy it down on a piece of
paper, and do not look at the solution in the book. Try working it, then compare your
results against the solution given, and, if necessary resolve, any differences. We have
tried to include most of the important steps in each example, but if something is not
clear you should always try—and here is where the pencil and paper come in again—
to fill in the details or missing steps. This may not be easy, but that is part of the learning process. The accumulation of facts followed by the slow assimilation of understanding simply cannot be achieved without a struggle.
Specifically for you, a Student Resource Manual (SRM) is available as an optional supplement. In addition to containing solutions of selected problems from the
exercises sets, the SRM contains hints for solving problems, extra examples, and a review of those areas of algebra and calculus that we feel are particularly important to
the successful study of differential equations. Bear in mind you do not have to purchase the SRM; by following my pointers given at the beginning of most sections, you
can review the appropriate mathematics from your old precalculus or calculus texts.
In conclusion, we wish you good luck and success. We hope you enjoy the text
and the course you are about to embark on—as undergraduate math majors it was
one of our favorites because we liked mathematics that connected with the physical
world. If you have any comments, or if you find any errors as you read/work your
way through the text, or if you come up with a good idea for improving either it or
the SRM, please feel free to contact us through our editor at Cengage Learning:
TO THE INSTRUCTOR
In case you are examining this book for the first time, Differential Equations with
Boundary-Value Problems, Eighth Edition can be used for either a one-semester course,
or a two-semester course that covers ordinary and partial differential equations. The
shorter version of the text, A First Course in Differential Equations with Modeling
Applications, Tenth Edition, is intended for either a one-semester or a one-quarter course
in ordinary differential equations. This book ends with Chapter 9. For a one semester
course, we assume that the students have successfully completed at least two semesters
of calculus. Since you are reading this, undoubtedly you have already examined the
Preface
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table of contents for the topics that are covered. You will not find a “suggested syllabus” in this preface; we will not pretend to be so wise as to tell other teachers what to
teach. We feel that there is plenty of material here to pick from and to form a course to
your liking. The textbook strikes a reasonable balance between the analytical, qualitative, and quantitative approaches to the study of differential equations. As far as our
“underlying philosophy” it is this: An undergraduate textbook should be written with
the student’s understanding kept firmly in mind, which means to me that the material
should be presented in a straightforward, readable, and helpful manner, while keeping
the level of theory consistent with the notion of a “first course.
For those who are familiar with the previous editions, we would like to mention
a few of the improvements made in this edition.
• Eight new projects appear at the beginning of the book. Each project includes
a related problem set, and a correlation of the project material with a chapter
in the text.
• Many exercise sets have been updated by the addition of new problems to
better test and challenge the students. In like manner, some exercise sets have
been improved by sending some problems into retirement.
• Additional examples and figures have been added to many sections
• Several instructors took the time to e-mail us expressing their concerns
about our approach to linear first-order differential equations. In response,
Section 2.3, Linear Equations, has been rewritten with the intent to simplify
the discussion.
• This edition contains a new section on Green’s functions in Chapter 4 for those
who have extra time in their course to consider this elegant application of
variation of parameters in the solution of initial-value and boundary-value problems. Section 4.8 is optional and its content does not impact any other section.
• Section 5.1 now includes a discussion on how to use both trigonometric
forms
in describing simple harmonic motion.
• At the request of users of the previous editions, a new section on the review
of power series has been added to Chapter 6. Moreover, much of this chapter
has been rewritten to improve clarity. In particular, the discussion of the
modified Bessel functions and the spherical Bessel functions in Section 6.4
has been greatly expanded.
• Several boundary-value problems involving modified Bessel functions have
been added to Exercises 13.2.
STUDENT RESOURCES
• Student Resource Manual (SRM), prepared by Warren S. Wright and Carol D.
Wright (ISBN 9781133491927 accompanies A First Course in Differential
Equations with Modeling Applications, Tenth Edition, and ISBN 9781133491958
accompanies Differential Equations with Boundary-Value Problems, Eighth
Edition), provides important review material from algebra and calculus, the
solution of every third problem in each exercise set (with the exception of the
Discussion Problems and Computer Lab Assignments), relevant command
syntax for the computer algebra systems Mathematica and Maple, lists of
important concepts, as well as helpful hints on how to start certain problems.
INSTRUCTOR RESOURCES
• Instructor’s Solutions Manual (ISM) prepared by Warren S. Wright and
Carol D. Wright (ISBN 9781133602293) provides complete, worked-out
solutions for all problems in the text.
• Solution Builder is an online instructor database that offers complete, workedout solutions for all exercises in the text, allowing you to create customized,
y Asin(vt f) and y Acos(vt f)
xii ● PREFACE
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PREFACE ● xiii
secure solutions printouts (in PDF format) matched exactly to the problems you
assign in class. Access is available via
www.cengage.com/solutionbuilder
• ExamView testing software allows instructors to quickly create, deliver, and
customize tests for class in print and online formats, and features automatic
grading. Included is a test bank with hundreds of questions customized directly to the text, with all questions also provided in PDF and Microsoft
Word formats for instructors who opt not to use the software component.
• Enhanced WebAssign is the most widely used homework system in higher
education. Available for this title, Enhanced WebAssign allows you to assign,
collect, grade, and record assignments via the Web. This proven homework
system includes links to textbook sections, video examples, and problem specific tutorials. Enhanced WebAssign is more than a homework system—it is
a complete learning system for students.
ACKNOWLEDGMENTS
We would like to single out a few people for special recognition. Many thanks to
Molly Taylor (senior sponsoring editor), Shaylin Walsh Hogan (assistant editor), and
Alex Gontar (editorial assistant) for orchestrating the development of this edition and
its component materials. Alison Eigel Zade (content project manager) offered the
resourcefulness, knowledge, and patience necessary to a seamless production process.
Ed Dionne (project manager, MPS) worked tirelessly to provide top-notch publishing
services. And finall , we thank Scott Brown for his superior skills as accuracy reviewer.
Once again an especially heartfelt thank you to Leslie Lahr, developmental editor, for
her support, sympathetic ear, willingness to communicate, suggestions, and for
obtaining and organizing the excellent projects that appear at the front of the book.
We also extend our sincerest appreciation to those individuals who took the time out
of their busy schedules to submit a project:
Ivan Kramer, University of Maryland—Baltimore County
Tom LaFaro, Gustavus Adolphus College
Jo Gascoigne, Fisheries Consultant
C. J. Knickerbocker, Sensis Corporation
Kevin Cooper, Washington State University
Gilbert N. Lewis, Michigan Technological University
Michael Olinick, Middlebury College
Finally, over the years these textbooks have been improved in a countless number of ways through the suggestions and criticisms of the reviewers. Thus it is fittin
to conclude with an acknowledgement of our debt to the following wonderful people
for sharing their expertise and experience.
REVIEWERS OF PAST EDITIONS
William Atherton, Cleveland State University
Philip Bacon, University of Florida
Bruce Bayly, University of Arizona
William H. Beyer, University of Akron
R. G. Bradshaw, Clarkson College
Dean R. Brown, Youngstown State University
David Buchthal, University of Akron
Nguyen P. Cac, University of Iowa
T. Chow, California State University—Sacramento
Dominic P. Clemence, North Carolina Agricultural and Technical
State University
Pasquale Condo, University of Massachusetts—Lowell
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