July 25, 2012 19:03 ffirs Sheet number 4 Page number iv cyan black WileyPLUS builds students’ confidence because it takes the guesswork out of studying by providing students with a clear roadmap: It offers interactive resources along with a complete digital textbook that help students learn more. With WileyPLUS , students take more initiative so you’ll have greater impact on their achievement in the classroom and beyond. ALL THE HELP, RESOURCES, AND PERSONAL SUPPORT YOU AND YOUR STUDENTS NEED! Technical Support 24/7 FAQs, online chat, and phone support www.wileyplus.com/support Student support from an experienced student user Collaborate with your colleagues, find a mentor, attend virtual and live events, and view resources 2-Minute Tutorials and all of the resources you and your students need to get started Your WileyPLUS Account Manager, providing personal training and support www.WhereFacultyConnect.com Pre-loaded, ready-to-use assignments and presentations created by subject matter experts July 25, 2012 19:03 ffirs Sheet number 3 Page number iii cyan black Elementary Differential Equations and Boundary Value Problems July 25, 2012 19:03 ffirs Sheet number 4 Page number iv cyan black July 25, 2012 19:03 ffirs Sheet number 5 Page number v cyan black T E N T H E D I T I O N Elementary Differential Equations and Boundary Value Problems William E. Boyce Edward P. Hamilton Professor Emeritus Richard C. DiPrima formerly Eliza Ricketts Foundation Professor Department of Mathematical Sciences Rensselaer Polytechnic Institute July 25, 2012 19:03 ffirs Sheet number 6 Page number vi cyan black PUBLISHER Laurie Rosatone ACQUISITIONS EDITOR David Dietz MARKETING MANAGER Melanie Kurkjian SENIOR EDITORIAL ASSISTANT Jacqueline Sinacori FREELANCE DEVELOPMENT EDITOR Anne Scanlan-Rohrer SENIOR PRODUCTION EDITOR Kerry Weinstein SENIOR CONTENT MANAGER Karoline Luciano SENIOR DESIGNER Madelyn Lesure SENIOR PRODUCT DESIGNER Tom Kulesa EDITORIAL OPERATIONS MANAGER Melissa Edwards ASSOCIATE CONTENT EDITOR Beth Pearson MEDIA SPECIALIST Laura Abrams ASSISTANT MEDIA EDITOR Courtney Welsh PRODUCTION SERVICES Carol Sawyer/The Perfect Proof COVER ART Norm Christiansen This book was set in Times Ten by MPS Limited, Chennai, India and printed and bound by R.R. Donnelley/ Willard. The cover was printed by R.R. Donnelley / Willard. This book is printed on acid free paper. ∞ The paper in this book was manufactured by a mill whose forest management programs include sustained yield harvesting of its timberlands. Sustained yield harvesting principles ensure that the numbers of trees cut each year does not exceed the amount of new growth. Copyright © 2012 John Wiley & Sons, Inc. All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, scanning, or otherwise, except as permitted under Sections 107 and 108 of the 1976 United States Copyright Act, without either the prior written permission of the Publisher, or authorization through payment of the appropriate per-copy fee to the Copyright Clearance Center, 222 Rosewood Drive, Danvers, MA 01923, (978) 750-8400, fax (978) 750-4470. Requests to the Publisher for permission should be addressed to the Permissions Department, John Wiley & Sons, Inc., 111 River Street, Hoboken, NJ 07030, (201) 748-6011, fax (201) 748-6008, E-mail: PERMREQ@WILEY.COM. To order books or for customer service, call 1 (800)-CALL-WILEY (225-5945). ISBN 978-0-470-45831-0 Printed in the United States of America 10 9 8 7 6 5 4 3 2 1 July 25, 2012 19:03 ffirs Sheet number 7 Page number vii cyan black To Elsa and in loving memory of Maureen To Siobhan, James, Richard, Jr., Carolyn, and Ann And to the next generation: Charles, Aidan, Stephanie, Veronica, and Deirdre July 25, 2012 19:03 ffirs Sheet number 8 Page number viii cyan black 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 Associ- ation of America, and the Society for Industrial and Applied Mathematics. He is currently the Edward P. Hamilton Distinguished Professor Emeritus of Science Ed- ucation (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 coau- thor (with R.L. Borrelli and C.S. Coleman) of Differential Equations Laboratory Workbook (Wiley 1992), which received the EDUCOM Best Mathematics Curricu- lar 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 inno- vation 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 Ex- ecutive Committee of the Applied Mechanics Division of ASME. In 1980, he was the recipient of the William H. Wiley Distinguished Faculty Award given by Rensselaer. He received Fulbright fellowships in 1964–65 and 1983 and a Guggenheim fellow- ship 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. July 20, 2012 18:17 fpref Sheet number 1 Page number ix cyan black ix P R E F A C E This edition, like its predecessors, is written from the viewpoint of the applied mathematician, whose interest in differential equations may be sometimes quite theoretical, sometimes intensely practical, and often somewhere in between. We have sought to combine a sound and accurate (but not abstract) exposition of the elementary theory of differential equations with considerable material on methods of solution, analysis, and approximation that have proved useful in a wide variety of applications. The book is written primarily for undergraduate students of mathematics, science, or engineering, who typically take a course on differential equations during their first or second year of study. The main prerequisite for reading the book is a working knowledge of calculus, gained from a normal two- or three-semester course sequence or its equivalent. Some familiarity with matrices will also be helpful in the chapters on systems of differential equations. To be widely useful, a textbook must be adaptable to a variety of instructional strategies. This implies at least two things. First, instructors should have maximum flexibility to choose both the particular topics they wish to cover and the order in which they want to cover them. Second, the book should be useful to students who have access to a wide range of technological capability. With respect to content, we provide this flexibility by making sure that, so far as possible, individual chapters are independent of each other. Thus, after the basic parts of the first three chapters are completed (roughly Sections 1.1 through 1.3, 2.1 through 2.5, and 3.1 through 3.5), the selection of additional topics, and the order and depth in which they are covered, are 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 that Chapter 10 should precede Chapter 11. This means that there are multiple pathways through the book, and many different combinations have been used effectively with earlier editions. July 20, 2012 18:17 fpref Sheet number 2 Page number x cyan black x Preface With respect to technology, we note repeatedly in the text that computers are ex- tremely useful for investigating differential equations and their solutions, and many of the problems are best approached with computational assistance. Nevertheless, the book is adaptable to courses having various levels of computer involvement, ranging from little or none to intensive. The text is independent of any particular hardware platform or software package. Many problems are marked with the symbol to indicate that we consider them to be technologically intensive. Computers have at least three important uses in a differential equations course. The first is simply to crunch numbers, thereby gen- erating accurate numerical approximations to solutions. The second is to carry out symbolic manipulations that would be tedious and time-consuming to do by hand. Finally, and perhaps most important of all, is the ability to translate the results of numerical or symbolic computations into graphical form, so that the behavior of solutions can be easily visualized. The marked problems typically involve one or more of these features. Naturally, the designation of a problem as technologically intensive is a somewhat subjective judgment, and the is intended only as a guide. Many of the marked problems can be solved, at least in part, without computa- tional help, and a computer can also be used effectively on many of the unmarked problems. From a student’s point of view, the problems that are assigned as homework and that appear on examinations drive the course. We believe that the most outstanding feature of this book is the number, and above all the variety and range, of the prob- lems 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. There are 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 motivation for solving many differential equations is the desire to learn some- thing about an underlying physical process that the equation is believed to model. It is basic to the importance of differential equations that even the simplest equa- tions correspond to useful physical models, such as exponential growth and decay, spring–mass systems, or electrical circuits. Gaining an understanding of a complex natural process is usually accomplished by combining or building upon simpler and more basic models. Thus a thorough knowledge of these basic models, the equations that describe them, and their solutions is the first and indispensable step toward the solution of more complex and realistic problems. We describe the modeling process in detail in Sections 1.1, 1.2, and 2.3. Careful constructions of models appear also in Sections 2.5 and 3.7 and in the appendices to Chapter 10. Differential equations re- sulting from the modeling process appear frequently throughout the book, especially in the problem sets. The main reason for including fairly extensive material on applications and math- ematical modeling in a book on differential equations is to persuade students that mathematical modeling often leads to differential equations, and that differential equations are part of an investigation of problems in a wide variety of other fields. We also emphasize the transportability of mathematical knowledge: once you mas- ter a particular solution method, you can use it in any field of application in which an appropriate differential equation arises. Once these points are convincingly made, we believe that it is unnecessary to provide specific applications of every method July 20, 2012 18:17 fpref Sheet number 3 Page number xi cyan black Preface xi of solution or type of equation that we consider. This helps to keep this book to a reasonable size, and in any case, there is only a limited time in most differential equations courses to discuss modeling and applications. Nonroutine problems often require the use of a variety of tools, both analytical and numerical. Paper-and-pencil methods must often be combined with effective use of a computer. Quantitative results and graphs, often produced by a computer, serve to illustrate and clarify conclusions that may be obscured by complicated ana- lytical expressions. On the other hand, the implementation of an efficient numerical procedure typically rests on a good deal of preliminary analysis—to determine the qualitative features of the solution as a guide to computation, to investigate limit- ing or special cases, or to discover which ranges of the variables or parameters may require or merit special attention. Thus, a student should come to realize that investi- gating a difficult problem may well require both analysis and computation; that good judgment may be required to determine which tool is best suited for a particular task; and that results can often be presented in a variety of forms. 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 a critical value 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 few or many of these problems. Readers familiar with the preceding edition will observe that the general structure of the book is unchanged. The 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. Sections 8.5 and 8.6 have been interchanged, so that the more advanced topics appear at the end of the chapter. 2. Derivations and proofs in several chapters have been expanded or rewritten to provide more details. 3. The fact that the real and imaginary parts of a complex solution of a real problem are also solutions now appears as a theorem in Sections 3.2 and 7.4. 4. The treatment of generalized eigenvectors in Section 7.8 has been expanded both in the text and in the problems. 5. There are about twenty new or revised problems scattered throughout the book. 6. There are new examples in Sections 2.1, 3.8, and 7.5. 7. About a dozen figures have been modified, mainly by using color to make the essen- tial feature of the figure more prominent. In addition, numerous captions have been July 20, 2012 18:17 fpref Sheet number 4 Page number xii cyan black xii Preface expanded to clarify the purpose of the figure without requiring a search of the surrounding text. 8. There are several new historical footnotes, and some others have been expanded. The authors have found differential equations to be a never-ending source of in- teresting, 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 Grafton, New York March 13, 2012 July 20, 2012 18:17 fpref Sheet number 5 Page number xiii cyan black Preface xiii Supplemental Resources for Instructors and Students An Instructor’s Solutions Manual, ISBN 978-0-470-45834-1, includes solutions for all problems not contained in the Student Solutions Manual. A Student Solutions Manual, ISBN 978-0-470-45833-4, includes solutions for se- lected problems in the text. A Book Companion Site, www.wiley.com/college/boyce, provides a wealth of re- sources 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 per- tinent sections in the text. • Mathematica, Maple, and M ATLAB 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 tradi- tional 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 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 equa- tions (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 M ATLAB , 3rd ed., 2012, ISBN 978-1-118-37680-5 • 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 teach- ing and learning. WileyPLUS builds students’ confidence because it takes the guesswork out of studying by providing students with a clear roadmap: what to do, how to do it, if they did it right. Students will take more initiative so you’ll have greater impact on their achievement in the classroom and beyond. 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 hyper- links to questions, definitions, and supplements for quicker and easier support. July 20, 2012 18:17 fpref Sheet number 6 Page number xiv cyan black xiv Preface • Guided Online (GO) Exercises, which prompt students to build solutions step- by-step. Rather than simply grading an exercise answer as wrong, GO problems show students precisely where they are making a mistake. • 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 assignments. Use them as is, or customize them to fit the needs of your classroom. • Interactive Demonstrations, based on figures from the text, which help reinforce and deepen understanding of the key concepts of differential equations. Use them in class or assign them as homework. Worksheets are provided to help guide and structure the experience of mastering these concepts. July 19, 2012 22:54 flast Sheet number 1 Page number xv cyan black xv A C K N O W L E D G M E N T S 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: Vincent Bonini, California Polytechnic State University, San Luis Obispo Fengxin Chen, University of Texas San Antonio Carmen Chicone, University of Missouri Matthew Fahy, Northern Arizona University Isaac Goldbring, University of California at Los Angeles Anton Gorodetski, University of California Irvine Mansoor Haider, North Carolina State University David Handron, Carnegie Mellon University Thalia D. Jeffres, Wichita State University Akhtar Khan, Rochester Institute of Technology Joseph Koebbe, Utah State University Ilya Kudish, Kettering University Tong Li, University of Iowa Wen-Xiu Ma, University of South Florida Aldo Manfroi, University of Illinois Urbana-Champaign Will Murray, California State University Long Beach Harold R. Parks, Oregon State University William Paulsen, Arkansas State University Shagi-Di Shih, University of Wyoming John Starrett, New Mexico Institute of Mining and Technology David S. Torain II, Hampton University George Yates, Youngstown State University Nung Kwan (Aaron) Yip, Purdue University Yue Zhao, University of Central Florida July 19, 2012 22:54 flast Sheet number 2 Page number xvi cyan black xvi Acknowledgments To my colleagues and students at Rensselaer, whose suggestions and reactions through the years have done much to sharpen my knowledge of differential equa- tions, as well as my ideas on how to present the subject. To those readers of the preceding edition who called errors or omissions to my attention. To Tamas Wiandt (Rochester Institute of Technology), who is primarily responsi- ble for the revision of the Instructor’s Solutions Manual and the Student Solutions Manual, and to Charles Haines (Rochester Institute of Technology), who assisted in this undertaking. To Tom Polaski (Winthrop University), who checked the answers in the back of the text and the Instructor’s Solutions Manual for accuracy. To David Ryeburn (Simon Fraser University), who carefully checked the entire manuscript and page proofs at least four times and is responsible for many corrections and clarifications. To Douglas Meade (University of South Carolina), who gave indispensable assis- tance in a variety of ways: by reading the entire manuscript at an early stage and offering numerous suggestions; by materially assisting in expanding the historical footnotes and updating the references; and by assuming the primary responsibility for checking the accuracy of the page proofs. 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. Finally, and most important, to my wife Elsa for discussing questions both math- ematical and stylistic, and above all for her unfailing support and encouragement during the revision process. In a very real sense, this book is a joint product. William E. Boyce July 19, 2012 22:53 ftoc Sheet number 1 Page number xvii cyan black xvii C O N T E N T S Chapter 1 Introduction 1 1.1 Some Basic Mathematical Models; Direction Fields 1 1.2 Solutions of Some Differential Equations 10 1.3 Classification of Differential Equations 19 1.4 Historical Remarks 26 Chapter 2 First Order Differential Equations 31 2.1 Linear Equations; Method of Integrating Factors 31 2.2 Separable Equations 42 2.3 Modeling with First Order Equations 51 2.4 Differences Between Linear and Nonlinear Equations 68 2.5 Autonomous Equations and Population Dynamics 78 2.6 Exact Equations and Integrating Factors 95 2.7 Numerical Approximations: Euler’s Method 102 2.8 The Existence and Uniqueness Theorem 112 2.9 First Order Difference Equations 122 Chapter 3 Second Order Linear Equations 137 3.1 Homogeneous Equations with Constant Coefficients 137 3.2 Solutions of Linear Homogeneous Equations; the Wronskian 145 3.3 Complex Roots of the Characteristic Equation 158 3.4 Repeated Roots; Reduction of Order 167 3.5 Nonhomogeneous Equations; Method of Undetermined Coefficients 175 3.6 Variation of Parameters 186 3.7 Mechanical and Electrical Vibrations 192 3.8 Forced Vibrations 207 Chapter 4 Higher Order Linear Equations 221 4.1 General Theory of n th Order Linear Equations 221 4.2 Homogeneous Equations with Constant Coefficients 228 4.3 The Method of Undetermined Coefficients 236 4.4 The Method of Variation of Parameters 241 Chapter 5 Series Solutions of Second Order Linear Equations 247 5.1 Review of Power Series 247 5.2 Series Solutions Near an Ordinary Point, Part I 254 July 19, 2012 22:53 ftoc Sheet number 2 Page number xviii cyan black xviii Contents 5.3 Series Solutions Near an Ordinary Point, Part II 265 5.4 Euler Equations; Regular Singular Points 272 5.5 Series Solutions Near a Regular Singular Point, Part I 282 5.6 Series Solutions Near a Regular Singular Point, Part II 288 5.7 Bessel’s Equation 296 Chapter 6 The Laplace Transform 309 6.1 Definition of the Laplace Transform 309 6.2 Solution of Initial Value Problems 317 6.3 Step Functions 327 6.4 Differential Equations with Discontinuous Forcing Functions 336 6.5 Impulse Functions 343 6.6 The Convolution Integral 350 Chapter 7 Systems of First Order Linear Equations 359 7.1 Introduction 359 7.2 Review of Matrices 368 7.3 Systems of Linear Algebraic Equations; Linear Independence, Eigenvalues, Eigenvectors 378 7.4 Basic Theory of Systems of First Order Linear Equations 390 7.5 Homogeneous Linear Systems with Constant Coefficients 396 7.6 Complex Eigenvalues 408 7.7 Fundamental Matrices 421 7.8 Repeated Eigenvalues 429 7.9 Nonhomogeneous Linear Systems 440 Chapter 8 Numerical Methods 451 8.1 The Euler or Tangent Line Method 451 8.2 Improvements on the Euler Method 462 8.3 The Runge–Kutta Method 468 8.4 Multistep Methods 472 8.5 Systems of First Order Equations 478 8.6 More on Errors; Stability 482 Chapter 9 Nonlinear Differential Equations and Stability 495 9.1 The Phase Plane: Linear Systems 495 9.2 Autonomous Systems and Stability 508 9.3 Locally Linear Systems 519 9.4 Competing Species 531 9.5 Predator–Prey Equations 544