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388445 Nagle ttl.qxd1/9/0811:53 AMPage 1INSTRUCTOR’SSOLUTIONS MANUALFUNDAMENTALS OF DIFFERENTIAL EQUATIONSSEVENTH EDITIONANDFUNDAMENTALS OF DIFFERENTIAL EQUATIONSAND BOUNDARY VALUE PROBLEMSFIFTH EDITIONR. Kent NagleUniversity of South FloridaEdward B. SaffVanderbilt UniversityA. David SniderUniversity of South Florida

388445 Nagle ttl.qxd1/9/0811:53 AMPage 2This work is protected by United States copyright laws and is provided solelyfor the use of instructors in teaching their courses and assessing studentlearning. Dissemination or sale of any part of this work (including on theWorld Wide Web) will destroy the integrity of the work and is not permitted. The work and materials from it should never be made available tostudents except by instructors using the accompanying text in theirclasses. All recipients of this work are expected to abide by theserestrictions and to honor the intended pedagogical purposes and the needs ofother instructors who rely on these materials.Reproduced by Pearson Addison-Wesley from electronic files supplied by the author.Copyright 2008 Pearson Education, Inc.Publishing as Pearson Addison-Wesley, 75 Arlington Street, Boston, MA 02116.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, or otherwise, without the prior written permission of the publisher.ISBN-13: 978-0-321-38844-5ISBN-10: 0-321-38844-5

ContentsNotes to the InstructorSoftware Supplements. . . . . . . . . . . . .Computer Labs. . . . . . . . . . . . . . . .Group Projects . . . . . . . . . . . . . . . . .Technical Writing Exercises . . . . . . . . . .Student Presentations . . . . . . . . . . . . .Homework Assignments . . . . . . . . . . . .Syllabus Suggestions . . . . . . . . . . . . . .Numerical, Graphical, and Qualitative MethodsEngineering/Physics Applications. . . . . .Biology/Ecology Applications . . . . . . . . . . .Supplemental Group Projects9Detailed Solutions & Answers to Even-Numbered ProblemsCHAPTER 1 IntroductionExercises 1.1 Detailed Solutions.Exercises 1.2 Detailed Solutions.Exercises 1.3 Detailed Solutions.Exercises 1.4 Detailed Solutions.Tables . . . . . . . . . . . . . . . .Figures. . . . . . . . . . . . . . .11122233457.CHAPTER 2 First Order Differential EquationsExercises 2.2 Detailed Solutions. . . . . . . . . . .Exercises 2.3 Detailed Solutions. . . . . . . . . . .Exercises 2.4 Detailed Solutions. . . . . . . . . . .Exercises 2.5 Detailed Solutions. . . . . . . . . . .Exercises 2.6 Detailed Solutions. . . . . . . . . . .Review Problems Answers. . . . . . . . . . . . . .Tables . . . . . . . . . . . . . . . . . . . . . . . . . .Figures. . . . . . . . . . . . . . . . . . . . . . . . .17.17171822252829.353541485661707171iii

CHAPTER 3Mathematical Models and Numerical MethodsInvolving First Order EquationsExercises 3.2 Detailed Solutions. . . . . . . . . . . . . . . . . . .Exercises 3.3 Detailed Solutions. . . . . . . . . . . . . . . . . . .Exercises 3.4 Detailed Solutions. . . . . . . . . . . . . . . . . . .Exercises 3.5 Answers . . . . . . . . . . . . . . . . . . . . . . . . .Exercises 3.6 Answers . . . . . . . . . . . . . . . . . . . . . . . . .Exercises 3.7 Answers . . . . . . . . . . . . . . . . . . . . . . . . .Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Figures. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .CHAPTER 4 Linear Second OrderExercises 4.1 Detailed Solutions. .Exercises 4.2 Detailed Solutions. .Exercises 4.3 Detailed Solutions. .Exercises 4.4 Detailed Solutions. .Exercises 4.5 Detailed Solutions. .Exercises 4.6 Detailed Solutions. .Exercises 4.7 Detailed Solutions. .Exercises 4.8 Detailed Solutions. .Exercises 4.9 Detailed Solutions. .Exercises 4.10 Detailed Solutions . .Review Problems Answers. . . . .Figures. . . . . . . . . . . . . . . .ivEquations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .CHAPTER 5 Introduction to SystemsExercises 5.2 Answers . . . . . . . . . .Exercises 5.3 Answers . . . . . . . . . .Exercises 5.4 Answers . . . . . . . . . .Exercises 5.5 Answers . . . . . . . . . .Exercises 5.6 Answers . . . . . . . . . .Exercises 5.7 Answers . . . . . . . . . .Exercises 5.8 Answers . . . . . . . . . .Review Problems Answers. . . . . . .Tables . . . . . . . . . . . . . . . . . . .Figures. . . . . . . . . . . . . . . . . .and. . . . . . . . . . . . . . . . . . . . .CHAPTER 6Exercises 6.1Exercises 6.2Exercises 6.3Exercises 6.4Linear Differential. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Theory of Higher-OrderAnswers . . . . . . . . . .Answers . . . . . . . . . .Answers . . . . . . . . . .Answers . . . . . . . . . .Phase. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Plane Analysis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Equations. . . . . . . . . . . . . . . . . . . . . . . . 3194194195

Review Problems Answers. . . . . . . . . . . . . . . . . . . . . . . . . . . . 196CHAPTER 7 Laplace TransformsExercises 7.2 Detailed Solutions.Exercises 7.3 Detailed Solutions.Exercises 7.4 Detailed Solutions.Exercises 7.5 Detailed Solutions.Exercises 7.6 Detailed Solutions.Exercises 7.7 Detailed Solutions.Exercises 7.8 Detailed Solutions.Exercises 7.9 Detailed Solutions.Review Problems Answers. . . .Figures. . . . . . . . . . . . . . .CHAPTER 8 Series Solutions of DifferentialExercises 8.1 Answers . . . . . . . . . . . . . .Exercises 8.2 Answers . . . . . . . . . . . . . .Exercises 8.3 Answers . . . . . . . . . . . . . .Exercises 8.4 Answers . . . . . . . . . . . . . .Exercises 8.5 Answers . . . . . . . . . . . . . .Exercises 8.6 Answers . . . . . . . . . . . . . .Exercises 8.7 Answers . . . . . . . . . . . . . .Exercises 8.8 Answers . . . . . . . . . . . . . .Review Problems Answers. . . . . . . . . . .Figures. . . . . . . . . . . . . . . . . . . . . .CHAPTER 9 Matrix MethodsExercises 9.1 Answers . . . . .Exercises 9.2 Answers . . . . .Exercises 9.3 Answers . . . . .Exercises 9.4 Answers . . . . .Exercises 9.5 Answers . . . . .Exercises 9.6 Answers . . . . .Exercises 9.7 Answers . . . . .Exercises 9.8 Answers . . . . .Review Problems Answers. .Figures. . . . . . . . . . . . .for Linear. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Equations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Systems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90CHAPTER 10 Partial Differential Equations291Exercises 10.2 Answers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291Exercises 10.3 Answers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291Exercises 10.4 Answers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292v

Exercises 10.5 AnswersExercises 10.6 AnswersExercises 10.7 Answers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295CHAPTER 11 EigenvalueExercises 11.2 Answers.Exercises 11.3 Answers.Exercises 11.4 Answers.Exercises 11.5 Answers.Exercises 11.6 Answers.Exercises 11.7 Answers.Exercises 11.8 Answers.Review Problems AnswersProblems and Sturm-Liouville. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .CHAPTER 12 Stability of AutonomousExercises 12.2 Answers. . . . . . . . . .Exercises 12.3 Answers. . . . . . . . . .Exercises 12.4 Answers. . . . . . . . . .Exercises 12.5 Answers. . . . . . . . . .Exercises 12.6 Answers. . . . . . . . . .Exercises 12.7 Answers. . . . . . . . . .Review Problems Answers. . . . . . . .Figures. . . . . . . . . . . . . . . . . . .Systems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .CHAPTER 13 Existence and UniquenessExercises 13.1 Answers. . . . . . . . . . .Exercises 13.2 Answers. . . . . . . . . . .Exercises 13.3 Answers. . . . . . . . . . .Exercises 13.4 Answers. . . . . . . . . . .Review Problems Answers. . . . . . . . .viTheory. . . . . . . . . . . . . . . . . . . . .Equations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 308309.317317317318318318

Notes to the InstructorOne goal in our writing has been to create flexible texts that afford the instructor a varietyof topics and make available to the student an abundance of practice problems and projects.We recommend that the instructor read the discussion given in the preface in order to gainan overview of the prerequisites, topics of emphasis, and general philosophy of the text.Software SupplementsInteractive Differential Equations CD-ROM: By Beverly West (Cornell University),Steven Strogatz (Cornell University), Jean Marie McDill (California Polytechnic State University – San Luis Obispo), John Cantwell (St. Louis University), and Hubert Hohn (Massachusetts College of Arts) is a popular software directly tied to the text that focuses on helpingstudents visualize concepts. Applications are drawn from engineering, physics, chemistry, andbiology. Runs on Windows or Macintosh and is included free with every book.Instructor’s MAPLE/MATHLAB/MATHEMATICA manual: By Thomas W. Polaski (Winthrop University), Bruno Welfert (Arizona State University), and Maurino Bautista(Rochester Institute of Technology). A collection of worksheets and projects to aid instructors in integrating computer algebra systems into their courses. Available via Addison-WesleyInstructor’s Resource Center.MATLAB Manual ISBN 13: 978-0-321-53015-8; ISBN 10: 0-321-53015-2MAPLE Manual ISBN 13: 978-0-321-38842-1; ISBN 10: 0-321-38842-9MATHEMATICA Manual ISBN 13: 978-0-321-52178-1; ISBN 10: 0-321-52178-1Computer LabsA computer lab in connection with a differential equations course can add a whole new dimension to the teaching and learning of differential equations. As more and more collegesand universities set up computer labs with software such as MAPLE, MATLAB, DERIVE,MATHEMATICA, PHASEPLANE, and MACMATH, there will be more opportunities to include a lab as part of the differential equations course. In our teaching and in our texts, wehave tried to provide a variety of exercises, problems, and projects that encourage the studentto use the computer to explore. Even one or two hours at a computer generating phase planediagrams can provide the students with a feeling of how they will use technology together1

with the theory to investigate real world problems. Furthermore, our experience is that theythoroughly enjoy these activities. Of course, the software, provided free with the texts, isespecially convenient for such labs.Group ProjectsAlthough the projects that appear at the end of the chapters in the text can be workedout by the conscientious student working alone, making them group projects adds a socialelement that encourages discussion and interactions that simulate a professional work placeatmosphere. Group sizes of 3 or 4 seem to be optimal. Moreover, requiring that each individualstudent separately write up the group’s solution as a formal technical report for grading bythe instructor also contributes to the professional flavor.Typically, our students each work on 3 or 4 projects per semester. If class time permits, oralpresentations by the groups can be scheduled and help to improve the communication skillsof the students.The role of the instructor is, of course, to help the students solve these elaborate problems ontheir own and to recommend additional reference material when appropriate.Some additional Group Projects are presented in this guide (see page 9).Technical Writi