Differential Equations and Boundary Value Problems

1 BOUNDARYAND BOUNDARYVALUE PROBLEMSVALUE PROBLEMSC omputing and ModelingFifth EditionC. Henry EdwardsDavid E. PenneyThe University of Georgiawith the assistance ofDavid CalvisBaldwin Wallace CollegeBoston Columbus Indianapolis New York San Francisco Upper Saddle RiverAmsterdam Cape Town Dubai London Madrid Milan Munich Paris Montr eal TorontoDehli Mexico City S ao Paulo Sydney Hong Kong Seoul Singapore Taipei :William HoffmanEditorial Assistant:Salena CashaProject Manager:Beth HoustonMarketing Manager:Jeff WeidenaarMarketing Assistant:Brooke SmithSenior Author Support/Technology Specialist:Joe VetereRights and Permissions Advisor:Aptara, Specialist:Carol MelvilleAssociate Director of Design:Andrea NixDesign Team Lead:Heather ScottText Design, Production Coordination, Composition:Dennis Kletzing, Kletzing Typesetting :George NicholsCover Design.

2 Studio MontageCover Image:Onne van der Wal/CorbisMany of the designations used by manufacturers and sellers to distinguish their products are claimedas trademarks. Where those designations appear in this book, and Pearson Education was aware of atrademark claim, the designations have been printed in initial caps or all of Congress Cataloging-in-Publication DataEdwards, C. H. (Charles Henry) Differential Equations and Boundary Value Problems : computing and modeling / C. HenryEdwards, David E. Penney, The University of Georgia, David Calvis, Baldwin Wallace College. --Fifth cmISBN 978-0-321-79698-1 (hardcover)1. Differential Equations . 2. Boundary Value Problems . I. Penney, David E. II. Calvis,David. III. 2015515'.35--dc232013040067 Copyrightc 2015, 2008, 2004 Pearson Education, rights reserved.

3 No part of this publication may be reproduced, stored in a retrieval system, or trans-mitted, in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise,without the prior written permission of the publisher. Printed in the United States of America. Forinformation on obtaining permission for use of material in this work, please submit a written request toPearson Education, Inc., Rights and Contracts Department, 501 Boylston Street, Suite 900, Boston, MA02116, fax your request to 617-671-3447, or e-mail at EB 18 17 16 15 10: 0-321-79698-5 ISBN 13: ModulesviPrefaceviiCHAPTER11 First-Order Differential Equations and Mathematical Models as General and Particular Solutions Fields and Solution Curves Equations and Applications First-Order Equations Methods and Exact Equations 57 CHAPTER22 Mathematical Models and Numerical Models Solutions and Stability Velocity Models Approximation: Euler s Method Closer Look at the Euler Method Runge Kutta Method 126 CHAPTER33 Linear Equations of Higher.

4 Second-Order Linear Equations Solutions of Linear Equations Equations with Constant Coefficients Vibrations Equations and Undetermined Coefficients Oscillations and Resonance Circuits Problems and Eigenvalues to Systems of Differential Systems and Applications Method of Elimination Methods for Systems 249 CHAPTER55 Linear Systems of Differential and Linear Systems eigenvalue Method for Homogeneous Systems Gallery of Solution Curves of Linear Systems Systems and Mechanical Applications eigenvalue Solutions Exponentials and Linear Systems Linear Systems 363 CHAPTER66 Nonlinear Systems and and the Phase Plane and Almost Linear Systems Models: Predators and Competitors Mechanical Systems in Dynamical Systems 426 CHAPTER77 Laplace Transform Transforms and Inverse Transforms of Initial Value Problems and Partial Fractions , Integrals, and Products of Transforms and Piecewise Continuous Input Functions and Delta Functions 484 CHAPTER88 Power Series and Review of Power Series Solutions Near Ordinary Points Singular Points of Frobenius.

5 The Exceptional Cases s Equation of Bessel Functions Series Methodsand Partial Differential Functions and Trigonometric Series Fourier Series and Convergence Sine and Cosine Series of Fourier Series Conduction and Separation of Variables Strings and the One-Dimensional Wave Equation Temperature and Laplace s Equation 625 CHAPTER1010 eigenvalue Methods and Boundary Value liouville Problems and Eigenfunction Expansions of Eigenfunction Series Periodic Solutions and Natural Frequencies Coordinate Problems Phenomena 681 References for Further Study698 Appendix: Existence and Uniqueness of Solutions701 Answers to Selected MODULESAPPLICATION MODULESThe modules listed here follow the indicated sections in the text. Most provide computing projects that illustrate thecontent of the corresponding text ,Mathematica, and MATLAB versions of these investigations areincluded in the Applications Manual that accompanies this Slope Fields andSolution Logistic Temperature Algebra Modeling of Population Euler s Euler Kutta Second-Order Solution Third-Order Solution Solution of Linear Variation of and Kepler s Laws of Algebra Solution of and Solution of Linear Calculation of Eigenvalues Phase Plane Vibrations of Eigenvalues and Matrix Exponential Variation of Plane Portraits and Plane Portraits of Almost Own Wildlife Conservation Rayleigh, van der Pol.

6 AndFitzHugh-Nagumo Algebra Transforms and of Initial Value and Resonance Computation of the Frobenius Series Exceptional Case by Reduction Equations and Modified Algebra Calculation of Series of Piecewise Smooth Eigenfunction Heat Flow Beams and Diving Functions and Heated is a textbook for the standard introductory Differential Equations coursetaken by science and engineering students. Its updated content reflects thewide availability of technical computing environments likeMaple,Mathematica,and MATLAB that now are used extensively by practicing engineers and traditional manual and symbolic methods are augmented with coverage alsoof qualitative and computer-based methods that employ numerical computation andgraphical visualization to develop greater conceptual understanding.

7 A bonus ofthis more comprehensive approach is accessibility to a wider range of more realisticapplications of Differential Features of This RevisionThis 5th edition is a comprehensive and wide-ranging addition to fine-tuning the exposition (both text and graphics) in numeroussections throughout the book, new applications have been inserted (including bio-logical), and we have exploited throughout the new interactive computer technologythat is now available to students on devices ranging from desktop and laptop com-puters to smart phones and graphing calculators. It also utilizes computer algebrasystems such asMathematica, Maple, and MATLAB as well as online web sitessuch as , with a single exception of a new section inserted in Chapter 5 (notedbelow), the classtested table of contents of the book remains unchanged.

8 Therefore,instructors notes and syllabi will not require revision to continue teaching with thisnew conspicuous feature of this edition is the insertion of about 80 new computer-generated figures, many of them illustrating how interactive computer applicationswith slider bars or touchpad controls can be used to change initial values or param-eters in a Differential equation, allowing the user to immediately see in real time theresulting changes in the structure of its illustrations of the various types of revision and updating exhibited inthis edition:New Interactive Technology and GraphicsNew figures inserted through-out illustrate the facility offered by modern computing technology platformsfor the user to interactively vary initial conditions and other parameters inreal time. Thus, using a mouse or touchpad, the initial point for an initialvalue problem can be dragged to a new location, and the corresponding solu-tion curve is automatically redrawn and dragged along with its initial instance, see the Sections (page 28) application module and (page148).

9 Using slider bars in an interactive graphic, the coefficients or other pa-rameters in a linear system can be varied, and the corresponding changes in itsdirection field and phase plane portrait are automatically shown; for instance, the application module for Section (page 319). The number of termsused from an infinite series solution of a Differential equation can be varied,and the resulting graphical change in the corresponding approximate solutionis shown immediately; see the Section application module (page 516).New ExpositionIn a number of sections, new text and graphics have beeninserted to enhance student understanding of the subject matter. For instance,see the treatments of separable Equations in Section (page 30), linear equa-tions in Section (page 45), isolated critical points in Sections (page372) and (page 383), and the new example in Section (page 618)showing a vibrating string with a momentary flat spot.

10 Examples and ac-companying graphics have been updated in Sections , , and toillustrate new graphing ContentThe single entirely new section for this edition is , which is devoted to the construction of a gallery of phase plane por-traits illustrating all the possible geometric behaviors of solutions of the 2-dimensional linear systemx0 DAx. In motivation and preparation for thedetailed study of eigenvalue -eigenvector methods in subsequent sections ofChapter 5 (which then follow in the same order as in the previous edi-tion), Section shows how the particular arrangements of eigenvalues andeigenvectors of the coefficient matrixAcorrespond to identifiable patterns fingerprints, so to speak in the phase plane portrait of the resulting gallery is shown in the two pages of phase plane portraits thatcomprise Figure (pages 315-316) at the end of the section.