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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.

2 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: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.

3 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. 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.

4 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.

5 Euler s Method Closer Look at the Euler Method Runge Kutta Method 126 CHAPTER33 Linear Equations of Higher : 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

6 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.

7 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.

8 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

9 Rayleigh, van der Pol, 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.

10 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.