Mathematics - Elementary Differential Equations

1 SEVENTHEDITIONE lementary DifferentialEquations and BoundaryValue ProblemsWilliam E. BoyceEdward P. Hamilton Professor EmeritusRichard C. DiPrimaformerly Eliza Ricketts Foundation ProfessorDepartment of Mathematical SciencesRensselaer Polytechnic InstituteJohn Wiley & Sons, YorkChichesterWeinheimBrisbaneTorontoSin gaporeASSOCIATE EDITORMary JohenkMARKETING MANAGERJ ulie Z. LindstromPRODUCTION EDITORKen SantorCOVER DESIGNM ichael JungINTERIOR DESIGNF earn Cutter DeVicq DeCumptichILLUSTRATION COORDINATORS igmund MalinowskiThis book was set in Times Roman by Eigentype Compositors, and printed and bound byVon Hoffmann Press, Inc. The cover was printed by Phoenix Color book is printed on acid-free paper. The paper in this book was manufactured by a mill whose forest management programs include sustainedyield harvesting of its timberlands. Sustained yield harvesting principles ensure that the numbers of treescut each year does not exceed the amount of new 2001 John Wiley & Sons, Inc.

2 All rights part of this publication may be reproduced, stored in a retrieval system or transmittedin any form or by any means, electronic, mechanical, photocopying, recording, scanningor otherwise, except as permitted under Sections 107 or 108 of the 1976 United StatesCopyright Act, without either the prior written permission of the Publisher, orauthorization through payment of the appropriate per-copy fee to the CopyrightClearance Center, 222 Rosewood Drive, Danvers, MA 01923, (508) 750-8400, fax(508) 750-4470. Requests to the Publisher for permission should be addressed to thePermissions Department, John Wiley & Sons, Inc., 605 Third Avenue, New York, NY10158-0012, (212) 850-6011, fax (212) 850-6008, E-Mail: of Congress Cataloging in Publication Data:Boyce, William Differential Equations and boundary value problems / William E. Boyce,Richard C. DiPrima 7th 0-471-31999-6 (cloth : alk. paper)1. Differential Equations . 2. Boundary value problems. I. DiPrima, Richard C.

3 II. TitleQA371 .B773 2000515 .35 dc2100-023752 Printed in the United States of America10987654321To Elsa and MaureenTo Siobhan, James, Richard, Jr., Carolyn, and AnnAnd to the next generation:Charles, Aidan, Stephanie, Veronica, and DeirdreThe AuthorsWilliam E. Boycereceived his degree in Mathematics from Rhodes College,and his and degrees in Mathematics from Carnegie-Mellon University. Heis a member of the American Mathematical Society, the Mathematical Associationof America, and the Society of Industrial and Applied Mathematics . He is currentlythe Edward P. Hamilton Distinguished Professor Emeritus of Science Education(Department of Mathematical Sciences) at Rensselaer. He is the author of numeroustechnical papers in boundary value problems and random Differential Equations andtheir applications. He is the author of several textbooks including two differentialequations texts, and is the coauthor (with Holmes, Ecker, and ) of a text on using Maple to explore Calculus.

4 He is also coauthor ( Borrelli and Coleman) ofDifferential Equations Laboratory Workbook(Wiley 1992), which received the EDUCOM Best Mathematics Curricular InnovationAward in 1993. Professor Boyce was a member of the NSF-sponsored CODEE(Consortium for Ordinary Differential Equations Experiments) that led to thewidely-acclaimedODE Architect. He has also been active in curriculum innovationand 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 theWilliam H. Wiley Distinguished Faculty Award given by C. DiPrima(deceased) received his , , and degrees inMathematics from Carnegie-Mellon University. He joined the faculty of RensselaerPolytechnic Institute after holding research positions at MIT, Harvard, and HughesAircraft. He held the Eliza Ricketts Foundation Professorship of Mathematics atRensselaer, was a fellow of the American Society of Mechanical Engineers, theAmerican Academy of Mechanics, and the American Physical Society.

5 He was alsoa member of the American Mathematical Society, the Mathematical Association ofAmerica, and the Society of Industrial and Applied Mathematics . He served as theChairman of the Department of Mathematical Sciences at Rensselaer, as President ofthe Society of Industrial and Applied Mathematics , and as Chairman of the ExecutiveCommittee of the Applied Mechanics Division of ASME. In 1980, he was the recip-ient of the William H. Wiley Distinguished Faculty Award given by Rensselaer. Hereceived Fulbright fellowships in 1964 65 and 1983 and a Guggenheim fellowship in1982 83. He was the author of numerous technical papers in hydrodynamic stabilityand lubrication theory and two texts on Differential Equations and boundary valueproblems. Professor DiPrima died on September 10, edition, like its predecessors, is written from the viewpoint of the applied mathe-matician, whose interest in Differential Equations may be highly theoretical, intenselypractical, or somewhere in between.

6 We have sought to combine a sound and accurate(but not abstract) exposition of the Elementary theory of Differential Equations withconsiderable material on methods of solution, analysis, and approximation that haveproved useful in a wide variety of book is written primarily for undergraduate students of Mathematics , science,or engineering, who typically take a course on Differential Equations during their firstor second year of study. The main prerequisite for reading the book is a workingknowledge of calculus, gained from a normal two- or three-semester course sequenceor its Changing Learning EnvironmentThe environment in which instructors teach, and students learn, Differential equationshas changed enormously in the past few years and continues to evolve at a rapid equipment of some kind, whether a graphing calculator, a notebook com-puter, or a desktop workstation is available to most students of Differential equipment makes it relatively easy to execute extended numerical calculations,to generate graphical displays of a very high quality, and, in many cases, to carry outcomplex symbolic manipulations.

7 A high-speed Internet connection offers an enormousrange of further fact that so many students now have these capabilities enables instructors, ifthey wish, to modify very substantially their presentation of the subject and theirexpectations of student performance. Not surprisingly, instructors have widely varyingopinions as to how a course on Differential Equations should be taught under thesecircumstances. Nevertheless, at many colleges and universities courses on differentialequations are becoming more visual, more quantitative, more project-oriented, and lessformula-centered than in the ModelingThe main reason for solving many Differential Equations is to try to learn somethingabout an underlying physical process that the equation is believed to model. It is basicto the importance of Differential Equations that even the simplest Equations correspondto useful physical models, such as exponential growth and decay, spring-mass systems,or electrical circuits.

8 Gaining an understanding of a complex natural process is usuallyaccomplished by combining or building upon simpler and more basic models. Thusa thorough knowledge of these models, the Equations that describe them, and theirsolutions, is the first and indispensable step toward the solution of more complex andrealistic difficult problems often require the use of a variety of tools, both analytical andnumerical. We believe strongly that pencil and paper methods must be combined witheffective use of a computer. Quantitative results and graphs, often produced by a com-puter, serve to illustrate and clarify conclusions that may be obscured by complicatedanalytical expressions. On the other hand, the implementation of an efficient numericalprocedure typically rests on a good deal of preliminary analysis to determine thequalitative features of the solution as a guide to computation, to investigate limiting orspecial cases, or to discover which ranges of the variables or parameters may requireor merit special , a student should come to realize that investigating a difficult problem maywell require both analysis and computation; that good judgment may be required todetermine which tool is best-suited for a particular task; and that results can often bepresented in a variety of Flexible ApproachTo be widely useful a textbook must be adaptable to a variety of instructional implies at least two things.

9 First, instructors should have maximum flexibility tochoose both the particular topics that they wish to cover and also the order in whichthey want to cover them. Second, the book should be useful to students having accessto a wide range of technological respect to content, we provide this flexibility by making sure that, so far aspossible, individual chapters are independent of each other. Thus, after the basic partsof the first three chapters are completed (roughly Sections through , , and through ) the selection of additional topics, and the order and depth inwhich they are covered, is at the discretion of the instructor. For example, while there isa good deal of material on applications of various kinds, especially in Chapters 2, 3, 9,and 10, most of this material appears in separate sections, so that an instructor can easilychoose which applications to include and which to omit. Alternatively, an instructorwho wishes to emphasize a systems approach to Differential Equations can take upChapter 7 (Linear Systems) and perhaps even chapter 9 (Nonlinear AutonomousSystems) immediately after chapter 2.

10 Or, while we present the basic theory of linearequations first in the context of a single second order equation ( chapter 3), manyinstructors have combined this material with the corresponding treatment of higherorder Equations ( chapter 4) or of linear systems ( chapter 7). Many other choices andPrefaceixcombinations are also possible and have been used effectively with earlier editions ofthis respect to technology, we note repeatedly in the text that computers are ex-tremely useful for investigating Differential Equations and their solutions, and manyof the problems are best approached with computational assistance. Nevertheless, thebook is adaptable to courses having various levels of computer involvement, rangingfrom little or none to intensive. The text is independent of any particular hardwareplatform or software package. More than 450 problems are marked with the symbol to indicate that we consider them to be technologically intensive. These problems maycall for a plot, or for substantial numerical computation, or for extensive symbolic ma-nipulation, or for some combination of these requirements.