Elementary differential equations 11th edition boyce

Differential Equations

and Boundary Value

Problems

B O Y C E | D I P R I M A | M E A D E

11th Edition

Elementary

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Elementary Differential Equations and Boundary Value Problems

Eleventh Edition

WILLIAM E. BOYCE

Edward P. Hamilton Professor Emeritus Department of Mathematical Sciences

Rensselaer Polytechnic Institute

RICHARD C. DIPRIMA

formerly Eliza Ricketts Foundation Professor Department of Mathematical Sciences

Rensselaer Polytechnic Institute

DOUGLAS B. MEADE

Department of Mathematics University of South Carolina - Columbia

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To Elsa, Betsy, and in loving memory of Maureen

To Siobhan, James, Richard Jr., Carolyn, Ann, Stuart,

Michael, Marybeth, and Bradley

And to the next generation:

Charles, Aidan, Stephanie, Veronica, and Deirdre

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The Authors

WILLIAM E. BOYCE received his B.A. degree in

Mathematics from Rhodes College, and his M.S. and Ph.D.

degrees in Mathematics from Carnegie-Mellon University.

He is a member of the American Mathematical Society,

the Mathematical Association of America, and the Society

for Industrial and Applied Mathematics. He is currently the

Edward P. Hamilton Distinguished Professor Emeritus of

Science Education (Department of Mathematical Sciences)

at Rensselaer. He is the author of numerous technical papers

in boundary value problems and random differential equations

and their applications. He is the author of several textbooks

including two differential equations texts, and is the coauthor

(with M.H. Holmes, J.G. Ecker, and W.L. Siegmann) of a text

on using Maple to explore Calculus. He is also coauthor (with

R.L. Borrelli and C.S. Coleman) of Differential Equations

Laboratory Workbook (Wiley 1992), which received the

EDUCOM Best Mathematics Curricular Innovation Award

in 1993. Professor Boyce was a member of the NSF-sponsored

CODEE (Consortium for Ordinary Differential Equations

Experiments) that led to the widely-acclaimed ODE Architect.

He has also been active in curriculum innovation and reform.

Among other things, he was the initiator of the “Computers

in Calculus” project at Rensselaer, partially supported by the

NSF. In 1991 he received the William H. Wiley Distinguished

Faculty Award given by Rensselaer.

RICHARD C. DIPRIMA (deceased) received his B.S., M.S.,

and Ph.D. degrees in Mathematics from Carnegie-Mellon

University. He joined the faculty of Rensselaer Polytechnic

Institute after holding research positions at MIT, Harvard,

and Hughes Aircraft. He held the Eliza Ricketts Foundation

Professorship of Mathematics at Rensselaer, was a fellow of

the American Society of Mechanical Engineers, the American

Academy of Mechanics, and the American Physical Society.

He was also a member of the American Mathematical Society,

the Mathematical Association of America, and the Society

for Industrial and Applied Mathematics. He served as the

Chairman of the Department of Mathematical Sciences at

Rensselaer, as President of the Society for Industrial and

Applied Mathematics, and as Chairman of the Executive

Committee of the Applied Mechanics Division of ASME.

In 1980, he was the recipient of the William H. Wiley

Distinguished Faculty Award given byRensselaer. He received

Fulbright fellowships in 1964--65 and 1983 and a Guggenheim

fellowship in 1982--83. He was the author of numerous

technical papers in hydrodynamic stability and lubrication

theory and two texts on differential equations and boundary

value problems. Professor DiPrima died on September 10,

1984.

DOUGLAS B. MEADE received B.S. degrees in

Mathematics and Computer Science from Bowling Green

State University, an M.S. in Applied Mathematics from

Carnegie Mellon University, and a Ph.D. in mathematics

from Carnegie Mellon University. After a two-year stint

at Purdue University, he joined the mathematics faculty at

the University of South Carolina, where he is currently an

Associate Professor of mathematics and the Associate Dean for

Instruction, Curriculum, and Assessment in the College of Arts

and Sciences. He is a member of the American Mathematical

Society, Mathematics Association of America, and Society

for Industrial and Applied Mathematics; in 2016 he was

named an ICTCM Fellow at the International Conference on

Technology in Collegiate Mathematics (ICTCM). His primary

research interests are in the numerical solution of partial

differential equations arising from wave propagation problems

in unbounded domains and from population models for

infectious diseases. He is also well-known for his educational

uses of computer algebra systems, particularly Maple.

These include Getting Started with Maple (with M. May,

C-K. Cheung, and G. E. Keough, Wiley, 2009, ISBN 978-0-

470-45554-8), Engineer’s Toolkit: Maple for Engineers (with

E. Bourkoff, Addison-Wesley, 1998, ISBN 0-8053-6445-5),

and numerous Maple supplements for numerous calculus,

linear algebra, and differential equations textbooks - including

previous editions of this book. He was a member of the

MathDL New Collections Working Group for Single Variable

Calculus, and chaired the Working Groups for Differential

Equations and Linear Algebra. The NSF is partially supporting

his work, together with Prof. Philip Yasskin (Texas A&M), on

the Maplets for Calculus project.

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Preface

As we have prepared an updated edition our first priorities

are to preserve, and to enhance, the qualities that have made

previous editions so successful. In particular, we adopt the

viewpoint of an applied mathematician with diverse interests

in differential equations, ranging from quite theoretical to

intensely practical--and usually a combination of both. Three

pillars of our presentation of the material are methods

of solution, analysis of solutions, and approximations of

solutions. Regardless of the specific viewpoint adopted, we

have sought to ensure the exposition is simultaneously correct

and complete, but not needlessly abstract.

The intended audience is undergraduate STEM students

whose degree program includes an introductory course in

differential equations during the first two years. The essential

prerequisite is a working knowledge of calculus, typically a

two- or three-semester course sequence or an equivalent.While

a basic familiarity with matrices is helpful, Sections 7.2 and 7.3

provide an overview of the essential linear algebra ideas needed

for the parts of the book that deal with systems of differential

equations (the remainder of Chapter 7, Section 8.5, and

Chapter 9).

A strength of this book is its appropriateness in a

wide variety of instructional settings. In particular, it allows

instructors flexibility in the selection of and the ordering of

topics and in the use of technology. The essential core material

is Chapter 1, Sections 2.1 through 2.5, and Sections 3.1 through

3.5. After completing these sections, the selection of additional

topics, and the order and depth of coverage are generally at

the discretion of the instructor. Chapters 4 through 11 are

essentially independent of each other, except that Chapter 7

should precede Chapter 9, and Chapter 10 should precede

Chapter 11.

A particularly appealing aspect of differential equations

is that even the simplest differential equations have a direct

correspondence to realistic physical phenomena: exponential

growth and decay, spring-mass systems, electrical circuits,

competitive species, and wave propagation. More complex

natural processes can often be understood by combining and

building upon simpler and more basic models. A thorough

knowledge of these basic models, the differential equations

that describe them, and their solutions--either explicit solutions

or qualitative properties of the solution--is the first and

indispensable step toward analyzing the solutions of more

complex and realistic problems. The modeling process is

detailed in Chapter 1 and Section 2.3. Careful constructions

of models appear also in Sections 2.5, 3.7, 9.4, 10.5, and

10.7 (and the appendices to Chap er 10). Various problem sets

throughout the book include problems that involve modeling

to formulate an appropriate differential equation, and then

to solve it or to determine some qualitative properties of its

solution. The primary purposes of these applied problems are

to provide students with hands-on experience in the derivation

of differential equations, and to convince them that differential

equations arise naturally in a wide variety of real-world

applications.

Another important concept emphasized repeatedly

throughout the book is the transportability of mathematical

knowledge. While a specific solution method applies to only a

particular class of differential equations, it can be used in any

application in which that particular type of differential equation

arises. Once this point is made in a convincing manner, we

believe that it is unnecessary to provide specific applications of

every method of solution or type of equation that we consider.

This decision helps to keep this book to a reasonable size, and

allows us to keep the primary emphasis on the development

of more solution methods for additional types of differential

equations.

From a student’s point of view, the problems that are

assigned as homework and that appear on examinations define

the course. We believe that the most outstanding feature of

this book is the number, and above all the variety and range,

of the problems that it contains. Many problems are entirely

straightforward, but many others are more challenging, and

some are fairly open-ended and can even serve as the basis

for independent student projects. The observant reader will

notice that there are fewer problems in this edition than in

previous editions; many of these problems remain available

to instructors via the WileyPlus course. The remaining 1600

problems are still far more problems than any instructor can

use in any given course, and this provides instructors with a

multitude of choices in tailoring their course to meet their own

goals and the needs of their students. The answers to almost all

of these problems can be found in the pages at the back of the

book; full solutions are in either the Student’s Solution Manual

or the Instructor’s Solution Manual.

While we make numerous references to the use of

technology, we do so without limiting instructor freedom to

use as much, or as little, technology as they desire. Appropriate

technologies include advanced graphing calculators (TI

Nspire), a spreadsheet (Excel), web-based resources (applets),

computer algebra systems, (Maple, Mathematica, Sage),

scientific computation systems (MATLAB), or traditional

programming (FORTRAN, Javascript, Python). Problems

marked with a G are ones we believe are best approached with

a graphical tool; those marked with a N are best solved with the

use of a numerical tool. Instructors should consider setting their

own policies, consistent with their interests and intents about

student use of technology when completing assigned problems.

Many problems in this book are best solved through

a combination of analytic, graphic, and numeric methods.

Pencil-and-paper methods are used to develop a model that

is best solved (or analyzed) using a symbolic or graphic

tool. The quantitative results and graphs, frequently produced

using computer-based resources, serve to illustrate and to

clarify conclusions that might not be readily apparent from

a complicated explicit solution formula. Conversely, the

vii

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viii PREFACE

implementation of an efficient numerical method to obtain

an approximate solution typically requires a good deal of

preliminary analysis--to determine qualitative features of the

solution as a guide to computation, to investigate limiting

or special cases, or to discover ranges of the variables or

parameters that require an appropriate combination of both

analytic and numeric computation. Good judgment may well

be required to determine the best choice of solution methods

in each particular case. Within this context we point out that

problems that request a “sketch” are generally intended to

be completed without the use of any technology (except your

writing device).

We believe that it is important for students to understand

that (except perhaps in courses on differential equations) the

goal of solving a differential equation is seldom simply to

obtain the solution. Rather, we seek the solution in order to

obtain insight into the behavior of the process that the equation

purports to model. In other words, the solution is not an end

in itself. Thus, we have included in the text a great many

problems, as well as some examples, that call for conclusions

to be drawn about the solution. Sometimes this takes the form

of finding the value of the independent variable at which the

solution has a certain property, or determining the long-term

behavior of the solution. Other problems ask for the effect of

variations in a parameter, or for the determination of all values

of a parameter at which the solution experiences a substantial

change. Such problems are typical of those that arise in the

applications of differential equations, and, depending on the

goals of the course, an instructor has the option of assigning as

few or as many of these problems as desired.

Readers familiar with the preceding edition will observe

that the general structure of the book is unchanged. The

minor revisions that we have made in this edition are in

many cases the result of suggestions from users of earlier

editions. The goals are to improve the clarity and readability of

our presentation of basic material about differential equations

and their applications. More specifically, the most important

revisions include the following:

1. Chapter 1 has been rewritten. Instead of a separate section

on the History of Differential Equations, this material

appears in three installments in the remaining three

section.

2. Additional words of explanation and/or more explicit

details in the steps in a derivation have been added

throughout each chapter. These are too numerous and

widespread to mention individually, but collectively they

should help to make the book more readable for many

students.

3. There are about forty new or revised problems scattered

throughout the book. The total number of problems has

been reduced by about 400 problems, which are still

available throughWileyPlus, leaving about 1600 problems

in print.

4. There are new examples in Sections 2.1, 3.8, and 7.5.

5. The majority (is this correct?) of the figures have been

redrawn, mainly by the use full color to allow for easier

identification of critical properties of the solution. In

addition, numerous captions have been expanded to clarify

the purpose of the figure without requiring a search of the

surrounding text.

6. There are several new references, and some others have

been updated.

The authors have found differential equations to be a

never-ending source of interesting, and sometimes surprising,

results and phenomena. We hope that users of this book, both

students and instructors, will share our enthusiasm for the

subject.

William E. Boyce and Douglas B. Meade

Watervliet, New York and Columbia, SC

29 August 2016

Supplemental Resources for

Instructors and Students

An Instructor’s Solutions Manual, ISBN 978-1-119-16976-5,

includes solutions for all problems not contained in the Student

Solutions Manual.

A Student Solutions Manual, ISBN 978-1-119-16975-8,

includes solutions for selected problems in the text.

A Book Companion Site, www.wiley.com/college/boyce,

provides a wealth of resources for students and instructors,

including

• PowerPoint slides of important definitions, examples, and

theorems from the book, as well as graphics for presentation

in lectures or for study and note taking.

• Chapter Review Sheets, which enable students to test their

knowledge of key concepts. For further review, diagnostic

feedback is provided that refers to pertinent sections in the

text.

• Mathematica, Maple, and MATLAB data files for selected

problems in the text providing opportunities for further

exploration of important concepts.

• Projects that deal with extended problems normally not

included among traditional topics in differential equations,

many involving applications from a variety of disciplines.

These vary in length and complexity, and they can be

assigned as individual homework or as group assignments.

A series of supplemental guidebooks, also published by John

Wiley & Sons, can be used with Boyce/DiPrima/Meade in

order to incorporate computing technologies into the course.

These books emphasize numerical methods and graphical

analysis, showing how these methods enable us to interpret

solutions of ordinary differential equations (ODEs) in the real

world. Separate guidebooks cover each of the three major

mathematical software formats, but the ODE subject matter is

the same in each.

• Hunt, Lipsman, Osborn, and Rosenberg, Differential

Equations with MATLAB , 3rd ed., 2012, ISBN 978-1-118-

37680-5

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PREFACE ix

• Hunt, Lardy, Lipsman, Osborn, and Rosenberg, Differential

Equations with Maple, 3rd ed., 2008, ISBN 978-0-471-

77317-7

• Hunt, Outing, Lipsman, Osborn, and Rosenberg,

Differential Equations with Mathematica, 3rd ed., 2009,

ISBN 978-0-471-77316-0

WileyPLUS WileyPLUS is an innovative, research-based online environment for effective teaching and learning.

WileyPLUS builds students’ confidence because it takes

the guesswork out of studying by providing students with a

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WileyPLUS, is loaded with all of the supplements above,

and it also features

• The E-book, which is an exact version of the print text

but also features hyperlinks to questions, definitions, and

supplements for quicker and easier support.

• Guided Online (GO) Exercises, which prompt students to

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an exercise answer as wrong, GO problems show students

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• Homework management tools, which enable instructors

easily to assign and grade questions, as well as to gauge

student comprehension.

• QuickStart pre-designed reading and homework assign

ments. Use them as is, or customize them to fit the needs of

your classroom.

Acknowledgments It is a pleasure to express my appreciation to the many people who

have generously assisted in various ways in the preparation of this

book.

To the individuals listed below, who reviewed the manuscript

and/or provided valuable suggestions for its improvement:

Irina Gheorghiciuc, Carnegie Mellon University

Bernard Brooks, Rochester Institute of Technology

James Moseley, West Virginia University

D. Glenn Lasseigne, Old Dominion University

Stephen Summers, University of Florida

Fabio Milner, Arizona State University

Mohamed Boudjelkha, Rensselaer Polytechnic Institute

Yuval Flicker, The Ohio State University

Y. Charles Li, University of Missouri, Columbia

Will Murray, California State University, Long Beach

Yue Zhao, University of Central Florida

Vladimir Shtelen, Rutgers University

Zhilan Feng, Purdue University

Mathew Johnson, University of Kansas

Bulent Tosun, University of Alabama

Juha Pohjanpelto, Oregon State University

Patricia Diute, Rochester Institute of Technology

Ning Ju, Oklahoma State University

Ian Christie, West Virginia University

Jonathan Rosenberg, University of Maryland

Irina Kogan, North Carolina State University

To our colleagues and students at Rensselaer and The University

of South Carolina, whose suggestions and reactions through the years

have donemuch to sharpen our knowledge of differential equations, as

well as our ideas on how to present the subject.

To those readers of the preceding edition who called errors or

omissions to our attention.

To Tom Polaski (Winthrop University), who is primarily

responsible for the revision of the Instructor’s Solutions Manual and the Student Solutions Manual.

To Mark McKibben (West Chester University), who checked the

answers in the back of the text and the Instructor’s Solutions Manual for accuracy, and carefully checked the entire manuscript.

To the editorial and production staff of John Wiley & Sons, who

have always been ready to offer assistance and have displayed the

highest standards of professionalism.

The last, but most important, people we want to thank are our

wives: Elsa, for discussing questions both mathematical and stylistic

and above all for her unfailing support and encouragement, and Betsy,

for her encouragement, patience and understanding.

WILLIAM E. BOYCE AND DOUGLAS B. MEADE

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Brief Contents

PREFACE vii

1 Introduction 1

2 First-Order Differential Equations 24

3 Second-Order Linear Differential Equations 103

4 Higher-Order Linear Differential Equations 169

5 Series Solutions of Second-Order Linear Equations 189

6 The Laplace Transform 241

7 Systems of First-Order Linear Equations 281

8 Numerical Methods 354

9 Nonlinear Differential Equations and Stability 388

10 Partial Differential Equations and Fourier Series 463

11 Boundary Value Problems and Sturm-Liouville Theory 529

ANSWERS TO PROBLEMS 573

INDEX 60

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Contents

PREFACE vii

1 Introduction 1

1.1 Some Basic Mathematical Models; Direction

Fields 1

1.2 Solutions of Some Differential Equations 9

1.3 Classification of Differential Equations 16

2 First-Order Differential Equations 24

2.1 Linear Differential Equations; Method of

Integrating Factors 24

2.2 Separable Differential Equations 33

2.3 Modeling with First-Order Differential

Equations 39

2.4 Differences Between Linear and Nonlinear

Differential Equations 51

2.5 Autonomous Differential Equations and

Population Dynamics 58

2.6 Exact Differential Equations and Integrating

Factors 70

2.7 Numerical Approximations: Euler’s Method 76

2.8 The Existence and Uniqueness Theorem 83

2.9 First-Order Difference Equations 91

3 Second-Order Linear Differential Equations 103

3.1 Homogeneous Differential Equations with

Constant Coefficients 103

3.2 Solutions of Linear Homogeneous Equations; the

Wronskian 110

3.3 Complex Roots of the Characteristic Equation

120

3.4 Repeated Roots; Reduction of Order 127

3.5 Nonhomogeneous Equations; Method of

Undetermined Coefficients 133

3.6 Variation of Parameters 142

3.7 Mechanical and Electrical Vibrations 147

3.8 Forced Periodic Vibrations 159

4 Higher-Order Linear Differential Equations 169

4.1 General Theory of nth Order Linear Differential

Equations 169

4.2 Homogeneous Differential Equations with

Constant Coefficients 174

4.3 The Method of Undetermined Coefficients 181

4.4 The Method of Variation of Parameters 185

5 Series Solutions of Second-Order Linear Equations 189

5.1 Review of Power Series 189

5.2 Series Solutions Near an Ordinary Point,

Part I 195

5.3 Series Solutions Near an Ordinary Point,

Part II 205

5.4 Euler Equations; Regular Singular Points 211

5.5 Series Solutions Near a Regular Singular Point,

Part I 219

5.6 Series Solutions Near a Regular Singular Point,

Part II 224

5.7 Bessel’s Equation 230

6 The Laplace Transform 241

6.1 Definition of the Laplace Transform 241

6.2 Solution of Initial Value Problems 248

6.3 Step Functions 257

6.4 Differential Equations with Discontinuous Forcing

Functions 264

6.5 Impulse Functions 270

6.6 The Convolution Integral 275

7 Systems of First-Order Linear Equations 281

7.1 Introduction 281

7.2 Matrices 286

7.3 Systems of Linear Algebraic Equations; Linear

Independence, Eigenvalues, Eigenvectors 295

7.4 Basic Theory of Systems of First-Order Linear

Equations 304

7.5 Homogeneous Linear Systems with Constant

Coefficients 309

7.6 Complex-Valued Eigenvalues 319

7.7 Fundamental Matrices 329

7.8 Repeated Eigenvalues 337

7.9 Nonhomogeneous Linear Systems 345

8 Numerical Methods 354

8.1 The Euler or Tangent Line Method 354

8.2 Improvements on the Euler Method 363

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xii CONTENTS

8.3 The Runge-Kutta Method 367

8.4 Multistep Methods 371

8.5 Systems of First-Order Equations 376

8.6 More on Errors; Stability 378

9 Nonlinear Differential Equations and Stability 388

9.1 The Phase Plane: Linear Systems 388

9.2 Autonomous Systems and Stability 398

9.3 Locally Linear Systems 407

9.4 Competing Species 417

9.5 Predator-Prey Equations 428

9.6 Liapunov’s Second Method 435

9.7 Periodic Solutions and Limit Cycles 444

9.8 Chaos and Strange Attractors: The Lorenz

Equations 454

10 Partial Differential Equations and Fourier Series 463

10.1 Two-Point Boundary Value Problems 463

10.2 Fourier Series 469

10.3 The Fourier Convergence Theorem 477

10.4 Even and Odd Functions 482

10.5 Separation of Variables; Heat Conduction

in a Rod 488

10.6 Other Heat Conduction Problems 496

10.7 The Wave Equation: Vibrations of an Elastic

String 504

10.8 Laplace's Equation 514

11 Boundary Value Problems and Sturm-Liouville Theory 529

11.1 The Occurrence of Two-Point Boundary Value

Problems 529

11.2 Sturm-Liouville Boundary Value Problems 535

11.3 NonhomogeneousBoundaryValueProblems 545

11.4 Singular Sturm-Liouville Problems 556

11.5 Further Remarks on the Method of Separation

of Variables: A Bessel Series Expansion 562

11.6 Series of Orthogonal Functions: Mean

Convergence 566

ANSWERS TO PROBLEMS 573

INDEX 60

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