Book - Elementary Differential Equations 9th edition PDF

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Elementary Differential Equations and Boundary Value Problems

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N I N T H

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EDITION

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

John Wiley & Sons, Inc.

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PUBLISHER ACQUISITIONS EDITOR MARKETING MANAGER ASSOCIATE EDITOR EDITORIAL ASSISTANT SENIOR PRODUCTION EDITOR SENIOR DESIGNER COVER DESIGN

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Laurie Rosatone David Dietz Jaclyn Elkins Michael Shroff/Will Art Pamela Lashbrook Ken Santor Kevin Murphy Norm Christiansen

This book was set in Times Ten by Techsetters, Inc., 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 © 2009 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: [email protected]. To order books or for customer service, call 1 (800)-CALL-WILEY (225-5945). ISBN 978-0-470-38334-6 Printed in the United States of America 10 9 8 7 6 5 4 3 2 1

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To Elsa and Maureen To Siobhan, James, Richard, Jr., Carolyn, and Ann 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 by Rensselaer. 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.

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P R E FAC E

This edition, like its predecessors, is written from the viewpoint of the applied mathematician, whose interest in differential equations may sometimes be 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 that they wish to cover and also the order in which they want to cover them. Second, the book should be useful to students having 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, is 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.

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Preface With respect to technology, we note repeatedly in the text that computers are extremely 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 generating 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 computational 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 problems that it contains. Many problems are entirely straightforward, but many others are more challenging, and some are fairly open-ended, and can 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 something 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 equations 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, 3.7, and in the appendices to Chapter 10. Differential equations resulting 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 mathematical 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 master 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 of solution or type of equation that we consider. This helps to keep this book to

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xi 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 analytical 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 limiting 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 investigating 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, one is interested in 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 a great many problems, as well as some examples in the text, that call for conclusions to be drawn about the solution. Sometimes this takes the form of asking for the value of the independent variable at which the solution has a certain property, or to determine 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 have several goals: to streamline the presentation in a few places, to make the presentation more visual by adding some new figures, and to improve the exposition by including several new or improved examples. More specifically, the most important changes are the following: 1. We have removed the discussion of linear dependence and independence from Chapter 3 (Second Order Linear Equations), where it is difficult to explain their importance, and introduced these concepts later in Chapter 4 (Higher Order Linear Equations) and in Chapter 7 (Linear Systems), where they appear more naturally. This results in a smaller block of theoretical material at the beginning of Chapter 3. Since not all courses cover Chapter 4, we also avoid using the words “linear dependence” and “linear independence” in Chapters 5 (Power Series Solutions) and 6 (Laplace Transforms). 2. Sections 5.4 (Regular Singular Points) and 5.5 (Euler Equations) from the eighth edition have now been combined into a single section, with Euler equations appearing first. They are then used as the prototype of equations having regular singular points, resulting in a somewhat briefer and more compact presentation. 3. Chapter 9 (Nonlinear Autonomous Systems) has several modifications. The concept of basins of attraction now appears earlier (in Section 9.2). This section also includes two new examples and three new figures with the goal of providing visual evidence that near a critical point nonlinear systems (usually) behave very much like linear systems. Such

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4. 5.

6.

7.

systems are now referred to as “locally linear.” In the following sections the Jacobian matrix is used more systematically to construct these linear approximations. There are also about 25 new problems in this chapter. In response to suggestions from several users, we have begun the discussion of forced linear oscillators in Section 3.8 with an example, rather than a general presentation. There are eight new problems on Euler equations (making a total of fifteen) in Sections 3.3 and 3.4. This will enable instructors to cover this topic, if they wish, even if Chapter 5 is not to be used. Euler equations appear in the text in Section 5.4. There are several revisions in Chapter 6 clarifying the integration of piecewise continuous functions, the essential uniqueness of the Laplace transform, and the use of the delta function. In addition, there is a new example, a new figure, and six new problems on the use of the unit step function to represent more complicated step functions. The list of 32 misce...