(Solution Manual) Fundamentals of Differential Equations and Boundary Value Problems (Featured Titles for Differential Equations)

Nagle,Saff,Snider, 6

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Author(s): Nagle, Saff, Snider

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Fundamentals of DE & Boundary Value 6ED & 8ED Edward B. Saff Arthur D.Snider Test Bank 2021 Boston San Francisco New York London Toronto Sydney Tokyo Singapore Madrid Mexico City Munich Paris Cape Town Hong Kong Montreal INSTRUCTOR ’S SOLUTIONS MANUAL VIKTOR MAYMESKUL University of South Florida FUNDAMENTALS OF DIFFERENTIAL EQUATIONS EIGHTH EDITION and FUNDAMENTALS OF DIFFERENTIAL EQUATIONS AND BOUNDARY VALUE PROBLEMS SIXTH EDITION R. Kent Nagle Edward B. Saff Vanderbilt University Arthur David Snider University of South Florida www.pearsonhighered.com The author and publisher of this book have used their best efforts in preparing this book. These efforts include the development, research, and testing of the theories and programs to determine their effectiveness. The author and publisher make no warranty of any kind, expressed or implied, with regard to these programs or the documentation contained in this book. The author and publisher shall not be liable in any event for incidental or consequential damages in connection with, or arising out of, the furnishing, performance, or use of these programs. Reproduced by Pearson from electronic files supplied by the author. Copyright © 2012, 2008, 2004 Pearson Education, Inc. Publishing as Pearson, 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. Printed in the United States of America. ISBN-13: 978-0-321-74835-5 ISBN-10: 0-321-74835-2 1 2 3 4 5 6 OPM 16 15 14 13 12 Contents Notes to the Instructor 1 Supplements ..................................... 1 Computer Labs ................................... 2 Group Projects .................................... 2 Technical Writing Exercises ............................. 2 Student Presentations ................................ 3 H o m e w o r kA s s i g n m e n t s ............................... 3S y l l a b u sS u g g e s t i o n s ................................. 4 Numerical, Graphical, and Qualitative Methods .................. 4 Engineering/Physics Applications ......................... 6 Biology/Ecology Applications ............................ 8 Supplemental Group Projects 10 Detailed Solutions & Answers to Even-Numbered Problems 17 CHAPTER 1 Introduction 17 Exercises 1.1 Detailed Solutions ......................... 1 7 Exercises 1.2 Detailed Solutions ......................... 1 8 Exercises 1.3 Detailed Solutions ......................... 2 3 Exercises 1.4 Detailed Solutions ......................... 2 5 T a b l e s ........................................ 2 9 F i g u r e s ....................................... 3 0 CHAPTER 2 First Order Differential Equations 35 Exercises 2.2 Detailed Solutions ......................... 3 5 Exercises 2.3 Detailed Solutions ......................... 4 3 Exercises 2.4 Detailed Solutions ......................... 5 1 Exercises 2.5 Detailed Solutions ......................... 5 9 Exercises 2.6 Detailed Solutions ......................... 6 5 Review Problems Answers ............................ 7 4 T a b l e s ........................................ 7 5F i g u r e s ....................................... 7 6 Copyright c/circlecopyrt2012 Pearson Education, Inc. Publishing as Addison-Wesley. iCHAPTER 3 Mathematical Models and Numerical Methods Involving First Order Equations 77 Exercises 3.2 Detailed Solutions ......................... 7 7 Exercises 3.3 Detailed Solutions ......................... 8 5 Exercises 3.4 Detailed Solutions ......................... 9 1 Exercises 3.5 Answers ............................... 1 0 0 Exercises 3.6 Answers ............................... 1 0 1 Exercises 3.7 Answers ............................... 1 0 1 T a b l e s ........................................ 1 0 3F i g u r e s ....................................... 1 0 4 CHAPTER 4 Linear Second Order Equations 105 Exercises 4.1 Detailed Solutions ......................... 1 0 5 Exercises 4.2 Detailed Solutions ......................... 1 0 7 Exercises 4.3 Detailed Solutions ......................... 1 1 5 Exercises 4.4 Detailed Solutions ......................... 1 2 3 Exercises 4.5 Detailed Solutions ......................... 1 2 8 Exercises 4.6 Detailed Solutions ......................... 1 4 0 Exercises 4.7 Detailed Solutions ......................... 1 4 7 Exercises 4.8 Detailed Solutions ......................... 1 6 1 Exercises 4.9 Detailed Solutions ......................... 1 6 4 Exercises 4.10 Detailed Solutions ......................... 1 7 1 Review Problems Answers ............................ 1 7 5 F i g u r e s ....................................... 1 7 7 CHAPTER 5 Introduction to Systems and Phase Plane Analysis 181 Exercises 5.2 Answers ............................... 1 8 1 Exercises 5.3 Answers ............................... 1 8 3 Exercises 5.4 Answers ............................... 1 8 4 Exercises 5.5 Answers ............................... 1 8 6 Exercises 5.6 Answers ............................... 1 8 6 Exercises 5.7 Answers ............................... 1 8 6 Exercises 5.8 Answers ............................... 1 8 7 Review Problems Answers ............................ 1 8 8 T a b l e s ........................................ 1 8 9F i g u r e s ....................................... 1 9 1 CHAPTER 6 Theory of Higher-Order Linear Differential Equations 197 Exercises 6.1 Answers ............................... 1 9 7 Exercises 6.2 Answers ............................... 1 9 8 Exercises 6.3 Answers ............................... 1 9 8 Exercises 6.4 Answers ............................... 1 9 9 ii Copyright c/circlecopyrt2012 Pearson Education, Inc. Publishing as Addison-Wesley.Review Problems Answers ............................ 2 0 0 CHAPTER 7 Laplace Transforms 201 Exercises 7.2 Detailed Solutions ......................... 2 0 1 Exercises 7.3 Detailed Solutions ......................... 2 0 5 Exercises 7.4 Detailed Solutions ......................... 2 1 1 Exercises 7.5 Detailed Solutions ......................... 2 1 9 Exercises 7.6 Detailed Solutions ......................... 2 2 8 Exercises 7.7 Detailed Solutions ......................... 2 4 2 Exercises 7.8 Detailed Solutions ......................... 2 5 0 Exercises 7.9 Detailed Solutions ......................... 2 5 5 Review Problems Answers ............................ 2 6 5 F i g u r e s ....................................... 2 6 6 CHAPTER 8 Series Solutions of Differential Equations 271 Exercises 8.1 Answers ............................... 2 7 1 Exercises 8.2 Answers ............................... 2 7 2 Exercises 8.3 Answers ............................... 2 7 3 Exercises 8.4 Answers ............................... 2 7 4 Exercises 8.5 Answers ............................... 2 7 5 Exercises 8.6 Answers ............................... 2 7 5 Exercises 8.7 Answers ............................... 2 7 7 Exercises 8.8 Answers ............................... 2 7 8 Review Problems Answers ............................ 2 7 9 F i g u r e s ....................................... 2 8 0 CHAPTER 9 Matrix Methods for Linear Systems 281 Exercises 9.1 Answers ............................... 2 8 1 Exercises 9.2 Answers ............................... 2 8 1 Exercises 9.3 Answers ............................... 2 8 2 Exercises 9.4 Answers ............................... 2 8 5 Exercises 9.5 Answers ............................... 2 8 7 Exercises 9.6 Answers ............................... 2 8 9 Exercises 9.7 Answers ............................... 2 9 0 Exercises 9.8 Answers ............................... 2 9 2 Review Problems Answers ............................ 2 9 3 F i g u r e s ....................................... 2 9 5 CHAPTER 10 Partial Differential Equations 297 Exercises 10.2 Answers .............................. 2 9 7 Exercises 10.3 Answers .............................. 2 9 7 Exercises 10.4 Answers .............................. 2 9 8 Copyright c/circlecopyrt2012 Pearson Education, Inc. Publishing as Addison-Wesley. iiiExercises 10.5 Answers .............................. 2 9 9 Exercises 10.6 Answers .............................. 3 0 0 Exercises 10.7 Answers .............................. 3 0 1 CHAPTER 11 Eigenvalue Problems and Sturm-Liouville Equations 303 Exercises 11.2 Answers .............................. 3 0 3 Exercises 11.3 Answers .............................. 3 0 4 Exercises 11.4 Answers .............................. 3 0 4 Exercises 11.5 Answers .............................. 3 0 5 Exercises 11.6 Answers .............................. 3 0 6 Exercises 11.7 Answers .............................. 3 0 8 Exercises 11.8 Answers .............................. 3 0 9 Review Problems Answers ............................ 3 0 9 CHAPTER 12 Stability of Autonomous Systems 311 Exercises 12.2 Answers .............................. 3 1 1 Exercises 12.3 Answers .............................. 3 1 1 Exercises 12.4 Answers .............................. 3 1 2 Exercises 12.5 Answers .............................. 3 1 2 Exercises 12.6 Answers .............................. 3 1 3 Exercises 12.7 Answers .............................. 3 1 3 Review Problems Answers ............................ 3 1 4 F i g u r e s ....................................... 3 1 5 CHAPTER 13 Existence and Uniqueness Theory 323 Exercises 13.1 Answers .............................. 3 2 3 Exercises 13.2 Answers .............................. 3 2 3 Exercises 13.3 Answers .............................. 3 2 4 Exercises 13.4 Answers .............................. 3 2 4 Review Problems Answers ............................ 3 2 4 Appendix A: Review of Integration Techniques 325 Exercises A Detailed Solutions .......................... 3 2 5 iv Copyright c/circlecopyrt2012 Pearson Education, Inc. Publishing as Addison-Wesley.Notes to the Instructor One goal in our writing has been to create flexible texts that afford the instructor a variety of 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 gain an overview of the prerequisites, topics of emphasis, and general philosophy of the text. Supplements Student’s Solutions Manual: By Viktor Maymeskul. Contains complete, worked-out solu- tions to most odd-numbered exercises, providing students with an excellent study tool. ISBN 13: 978-0-321-74834-8; ISBN 10: 0-321-74834-4. Companion Web site: Provides additional resources for both instructors and students, includinghelpfullinkskeyedtosectionsofthetext,accesstoInteractiveDifferentialEquations, suggestions for incorporating Interactive Differential Equations modules, suggested syllabi, index of applications, and study tips for students. Access: www.pearsonhighered.com/nagle Interactive Differential Equations: ByBeverlyWest(CornellUniversity),StevenStrogatz (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 helping students visualize concepts. Applications are drawn from engineering, physics, chemistry, and biology. Access: www.pearsonhighered.com/nagle Instructor’s MAPLE/MATHLAB/MATHEMATICA manuals: By Thomas W. Po- laski (Winthrop University), Bruno Welfert (Arizona State University), and Maurino Bautista (Rochester Institute of Technology). A collection of worksheets and projects to aid instructors inintegratingcomputeralgebrasystemsintotheircourses. AvailableinthePearsonInstructor Resource Center at www.pearsonhighered.com/irc . MATLAB Manual ISBN 13: 978-0-321-53015-8; ISBN 10: 0-321-53015-2 MAPLE Manual ISBN 13: 978-0-321-38842-1; ISBN 10: 0-321-38842-9 MATHEMATICA Manual ISBN 13: 978-0-321-52178-1; ISBN 10: 0-321-52178-1 Copyright c/circlecopyrt2012 Pearson Education, Inc. Publishing as Addison-Wesley. 1Computer Labs A computer lab in connection with a differential equations course can add a whole new di- mension to the teaching and learning of differential equations. As more and more colleges and universities set up computer labs with software such as MAPLE, MATLAB, DERIVE, MATHEMATICA, PHASEPLANE, and MACMATH, there will be more opportunities to in- clude a lab as part of the differential equations course. In our teaching and in our texts, we have tried to provide a variety of exercises, problems, and projects that encourage the student to use the computer to explore. Even one or two hours at a computer generating phase plane diagrams can provide the students with a feeling of how they will use technology together with the theory to investigate real world problems. Furthermore, our experience is that they thoroughly enjoy these activities. Of course, the software, provided free with the texts, is especially convenient for such labs. Group Projects Although the projects that appear at the end of the chapters in the text can be workedout by the conscientious student working alone, making them groupprojects adds a social element that encourages discussion and interactions that simulate a professional work place atmosphere. Groupsizesof3or4seemtobeoptimal. Moreover, requiringthateachindividual student separately write up the group’s solution as a formal technical report for grading by the instructor also contributes to the professional flavor. Typically, our students each work on 3 or 4 projects per semester. If class time permits, oral presentations by the groups can be scheduled and help to improve the communication skills of the students. The role of the instructor is, of course, to help the students solve these elaborate problems on their own and to recommend additional reference material when appropriate. Some additional Group Projects are presented in this guide (see page 10). Technical Writing Exercises Thetechnicalwritingexercisesattheendofmostchaptersinvitestudentstomakedocumented responses to questions dealing with the concepts in the chapter. This not only gives students an opportunity to improve their writing skills, but it helps them organize their thoughts and better understand the new concepts. Moreover, many questions deal with critical thinking 2 Copyright c/circlecopyrt2012 Pearson Education, Inc. Publishing as Addison-Wesley.skills that will be useful in their careers as engineers, scientists, or mathematicians. Since most students have little experience with technical writing, it may be necessary to return ungraded thefirstfewtechnicalwritingassignmentswithcommentsandhavethestudentsredo the the exercise. This has worked well in our classes and is much appreciated by the students. Handing out a “model” technical writing response is also helpful for the students. Student Presentations It is not uncommon for an instructor to have students go to the board and present a solutionto a problem. Differential equations is so rich in theory and applications that it is an excellent course to allow (require) a student to give a presentation on a special application (e.g., almost any topic from Chapters 3 and 5), on a new technique not covered in class (e.g., material from Section 2.6, Projects A, B, or C in Chapter 4), or on additional theory (e.g., material from Chapter 6 which generalizes the results in Chapter 4). In addition to improving students’ communication skills, these “special” topics are long remembered by the students. Here, too, working in groups of 3 or 4 and sharing the presentation responsibilities can add substantially to the interest and quality of the presentation. Students should also be encouraged to enliventheir communication by building physical models, preparing part of their lectures with the aid of video technology, and utilizing appropriate internet web sites. Homework Assignments We would like to share with you an obvious, non-original, but effective method to encourage students to do homework problems. An essential feature is that it requires little extra work on the part of the instructor or grader. We assign homework problems (about 5 of them) after each lecture. At the end of the week (Fridays), students are asked to turn in their homework (typically, 3 sets) for that week. We then choose at random one problem from each assignment (typically, a total of 3) that will be graded. (The point is that the student does not know in advance which problems will be chosen.) Full credit is given for any of the chosen problems for which there is evidence that the student has made an honest attempt at solving. The homework problem sets are returned to the students at the next meeting (Mondays) with grades like 0 /3, 1/3, 2/3, or 3/3 indicating the proportion of problems for which the student received credit. The homework grades are tallied at the end of the semester and count as one test grade. Certainly, there are variations Copyright c/circlecopyrt2012 Pearson Education, Inc. Publishing as Addison-Wesley. 3on this theme. The point is that students are motivated to do their homework. Syllabus Suggestions To serve as a guide in constructing a syllabus for a one-semester or two-semester course, the prefaces to the texts list sample outlines that emphasize methods, applications, theory, partial differential equations, phase plane analysis, computation, or combinations of these. As a further guide in making a choice of subject matter, we provide (starting on the next page) a listing of text material dealing with some common areas of emphasis. Numerical, Graphical, and Qualitative Methods The sections and projects dealing with numerical, graphical, and qualitative techniques ofsolving differential equations include: Section 1.3: Direction Fields Section 1.4: The Approximation Method of Euler Project A for Chapter 1: Taylor Series Method Project B for Chapter 1: Picard’s Method Project C for Chapter 1: The Phase Line Section 3.6: Improved Euler’s Method , which includes step-by-step outlines of the im- proved Euler’s method subroutine and improved Euler’s method with tolerance. These outlines are easy for the student to translate into a computer program (pp. 127–128). Section 3.7: Higher-Order Numerical Methods :Taylor and Runge-Kutta , which includes outlines for the Fourth Order Runge-Kutta subroutine and algorithm with tolerance (see pp. 135–136). Project H for Chapter 3: Stability of Numerical Methods Project I for Chapter 3: Period Doubling and Chaos 4 Copyright c/circlecopyrt2012 Pearson Education, Inc. Publishing as Addison-Wesley.Section 4.8: Qualitative Considerations for Variable Coefficient and Non-linear Equa- tions, which discusses the energy integral lemma, as well as the Airy, Bessel, Duffing, and van der Pol equations. Section 5.3: Solving Systems and Higher-Order Equations Numerically , which describes the vectorized forms of Euler’s method and the Fourth Order Runge-Kutta method, and discusses an application to population dynamics. Section 5.4: Introduction to the Phase Plane , which introduces the study of trajectories of autonomous systems, critical points, and stability. Section 5.8: Dynamical Systems, Poincar´ e Maps, and Chaos , which discusses the use of numerical methods to approximate the Poincar` e map and how to interpret the results. Project A for Chapter 6: Computer Algebra Systems and Exponential Shift Project D for Chapter 6: Higher-Order Difference Equations Project A for Chapter 8: Alphabetization Algorithms Project D for Chapter 10: Numerical Method for Δu=fon a Rectangle Project D for Chapter 11: Shooting Method Project E for Chapter 11: Finite-Difference Method for Boundary Value Problems Project C for Chapter 12: Computing Phase Plane Diagrams Project D for Chapter 12: Ecosystem of Planet GLIA-2 Section 13.1: Introduction: Successive Approximations Appendix B: Newton’s Method Appendix C: Simpson’s Rule Appendix E: Method of Least Squares Appendix F: Runge-Kutta Pr ocedure for Equations Copyright c/circlecopyrt2012 Pearson Education, Inc. Publishing as Addison-Wesley. 5The instructor who wishes to emphasize numerical methods should also note that the text contains an extensive chapter of series solutions of differential equations (Chapter 8). Engineering/Physics Applications Since Laplace transforms is a subject vital to engineering, we have included a detailed chapter on this topic – see Chapter 7. Stability is also an important subject for engineers, so we have included an introduction to the subject in Section 5.4 along with an entire chapter addressing this topic – see Chapter 12. Further material dealing with engineering/physics applications include: Project C for Chapter 2: Torricelli’s Law of Fluid Flow . Project I for Chapter 2: Designing a Solar Collector . Section 3.1: Mathematical Modeling . Section 3.2: Compartmental Analysis , which contains a discussion of mixing problems and of population models. Section 3.3: Heating and Cooling off Buildings , which discusses temperature variations in the presence of air conditioning or furnace heating. Section 3.4: Newtonian Mechanics . Section 3.5: Electrical Circuits . Project C for Chapter 3: Curve of Pursuit . Project D for Chapter 3: Aircraft Guidance in a Crosswind . Project E for Chapter 3: Feedback and the Op Amp . Project F for Chapter 3: Bang-Bang Controls . Section 4.1: Introduction: The Mass-Spring Oscillator . Section 4.8: Qualitative Considerations for Variable-Coefficient and Non-linear Equa- tions. Section 4.9: A Closer Look at Free Mechanical Vibrations . 6 Copyright c/circlecopyrt2012 Pearson Education, Inc. Publishing as Addison-Wesley.Section 4.10: A Closer Look at Forced Mechanical Vibrations . Project B for Chapter 4: Apollo Re-entry Project C for Chapter 4: Simple Pendulum Section 5.1: Interconnected Fluid Tanks . Section 5.4: Introduction to the Phase PLane . Section 5.6: Coupled Mass-Spring Systems . Section 5.7: Electrical Systems . Section 5.8: Dynamical Systems, Poincar´ e Maps, and Chaos . Project A for Chapter 5: Designing a Landing System for Interplanetary Travel . Project C for Chapter 5: Things that Bob . Project D for Chapter 5: Hamiltonian Systems . Project C for Chapter 6: Transverse Vibrations of a Beam . Chapter 7: Laplace Transforms , which in addition to basic material includes discussions of transfer functions, the Dirac delta function, and frequency response modelling. Project B for Chapter 8, Spherically Symmetric Solutions to Schr¨ odinger’s Equation for the Hydrogen Atom Project D for Chapter 8, Buckling of a Tower Project E for Chapter 8, Aging Spring and Bessel Functions Section 9.6: Complex Eigenvalues , includes discussion of normal (natural) frequencies. Project B for Chapter 9: Matrix Laplace Transform Method . Project C for Chapter 9: Undamped Second-Order Systems . Chapter 10: Partial Differential Equations , which includes sections on Fourier series, the heat equation, wave equation, and Laplace’s equation. Copyright c/circlecopyrt2012 Pearson Education, Inc. Publishing as Addison-Wesley. 7Project A for Chapter 10: Steady-State Temperature Distribution in a Circular Cylinder . Project B for Chapter 10: A Laplace Transform Solution of the Wave Equation . Project A for Chapter 11: Hermite Polynomials and the Harmonic Oscillator . Section 12.4: Energy Methods , which addresses both conservative and non-conservative autonomous mechanical systems. Project A for Chapter 12: Solitons and Korteweg-de Vries Equation . Project B for Chapter 12: Burger’s Equation . Students of engineering and physics would also find Chapter 8 on series solutions particularly useful, especially Section 8.8 on special functions. Biology/Ecology Applications Project Cfor Chapter 1: The Phase Plane , which discussesthe logisticpopulation model and bifurcation diagrams for population control. Project A for Chapter 2: Oil Spill in a Canal . Project B for Chapter 2: Differential Equations in Clinical Medicine . Section 3.1: Mathematical Modelling . Section 3.2: Compartmental Analysis , which contains a discussion of mixing problems and population models. Project A for Chapter 3: Dynamics of HIV Infection . ProjectBforChapter3: Aquaculture , whichdealswithamodelofraisingandharvesting catfish. Section 5.1: Interconnected Fluid Tanks , which introduces systems of equations. Section 5.3: Solving Systems and Higher-Order Equations Numerically , which contains an application to population dynamics. 8 Copyright c/circlecopyrt2012 Pearson Education, Inc. Publishing as Addison-Wesley.Section 5.5: Applications to Biomathematics: Epidemic and Tumor Growth Models . Project B for Chapter 5: Spread of Staph Infections in Hospitals – Part I . Project E for Chapter 5: Cleaning Up the Great Lakes Project F for Chapter 5: A Growth Model for Phytoplankton – Part I . Problem 19 in Exercises 10.5, which involves chemical diffusion through a thin layer.Project D for Chapter 12: Ecosystem on Planet GLIA-2 Project E for Chapter 12: Spread of Staph Infections in Hospitals – Part II . Project F for Chapter 12: A Growth Model for Phytoplankton – Part II . The basic content of the remainder of this instructor’s manual consists of supplemental group projects, answers to the even-numbered problems, and detailed solutions to the most even- numbered problems in Chapters 1, 2, 3, 4, and 7. These answers are not available any place else since the text and the Student’s Solutions Manual only provide answers and solutions to odd-numbered problems. We would appreciate any comments you may have concerning the answers in this manual. These comments can be sent to the authors’ email addresses below. We also would encourage sharing with us (the authors and users of the texts) any of your favorite group projects. E. B. Saff A. D. Snider Edward.B.Saff@Vanderbilt.edu snider@eng.usf.edu Copyright c/circlecopyrt2012 Pearson Education, Inc. Publishing as Addison-Wesley. 9Group Projects for Chapter 3 Delay Differential Equations In our discussion of mixing problems in Section 3.2, we encountered the initial value problem x/prime(t)=6−3 500x(t−t0), (0.1) x(t)=0 f o r x∈[−t0,0], wheret0is a positive constant. The equation in (0.1) is an example of a delay differ- ential equation . These equations differ from the usual differential equations by the presence of the shift ( t−t0) in the argument of the unknown function x(t). In general, these equations are more difficult to work with than are regular differential equations, but quite a bit is known about them.1 (a)Show that the simple linear delay differential equation x/prime=ax(t−b), (0.2) wherea,bare constants, has a solution of the form x(t)=Cestfor any constant C, provided ssatisfies the transcendental equation s=ae−bs. (b)A solution to (0.2) for t>0 can also be found using the method of steps . Assume thatx(t)=f(t)f o r−b≤t≤0. For 0≤t≤b, equation (0.2) becomes x/prime(t)=ax(t−b)=af(t−b), and so x(t)=t⎬integraldisplay 0af(ν−b)dν+x(0). Now that we know x(t)o n[ 0,b], we can repeat this procedure to obtain x(t)=t⎬integraldisplay bax(ν−b)dν+x(b) forb≤x≤2b. This process can be continued indefinitely. 1See, for example, Differential–Difference Equations , by R. Bellman and K. L. Cooke, Academic Press, New York, 1963, or Ordinary and Delay Differential Equations , by R. D. Driver, Springer–Verlag, New York, 1977 10 Copyright c/circlecopyrt2012 Pearson Education, Inc. Publishing as Addison-Wesley.Use the method of steps to show that the solution to the initial value problem x/prime(t)=−x(t−1),x(t)=1 o n [ −1,0], is given by x(t)=n⎬summationdisplay k=0(−1)k[t−(k−1)]k k!,forn−1≤t≤n, wherenis a nonnegative integer. (This problem can also be solved using the Laplace transform method of Chapter 7.) (c)Use the method of steps to compute the solution to the initial value problem given in (0.1) on the interval 0 ≤t≤15 fort0=3 . Extrapolation When precise information about the formof the error in an approximation is known, a technique called extrapolation can be used to improve the rate of convergence. SupposetheapproximationmethodconvergeswithrateO( hp)ash→0(cf.Section3.6). From theoretical considerations, assume we know, more precisely, that y(x;h)=φ(x)+hpap(x)+O⎬parenleftbig hp+1⎬parenrightbig , (0.3) wherey(x;h) is the approximation to φ(x) using step size handap(x) is some function that is independent of h(typically, we do not know a formula for ap(x), only that it exists). Our goal is to obtain approximations that converge at the faster rate than O(hp+1). We start by replacing hbyh/2 in (0.3) to get y⎬parenleftbigg x;h 2⎬parenrightbigg =φ(x)+hp 2pap(x)+O⎬parenleftbig hp+1⎬parenrightbig . If we multiply both sides by 2pand subtract equation (0.3), we find 2py⎬parenleftbigg x;h 2⎬parenrightbigg −y(x;h)=( 2p−1)φ(x)+O⎬parenleftbig hp+1⎬parenrightbig . Solving for φ(x) yields φ(x)=2py(x;h/2)−y(x;h) 2p−1+O⎬parenleftbig hp+1⎬parenrightbig . Copyright c/circlecopyrt2012 Pearson Education, Inc. Publishing as Addison-Wesley. 11Hence, y∗⎬parenleftbigg x;h 2⎬parenrightbigg :=2py(x;h/2)−y(x;h) 2p−1 has a rate of convergence of O( hp+1). (a)Assuming y∗⎬parenleftbigg x;h 2⎬parenrightbigg =φ(x)+hp+1ap+1(x)+O⎬parenleftbig hp+2⎬parenrightbig , show that y∗∗⎬parenleftbigg x;h 4⎬parenrightbigg :=2p+1y∗(x;h/4)−y∗(x;h/2) 2p+1−1 h a sar a t eo fc o n v e r g e n c eo fO( hp+2). (b)Assuming y∗∗⎬parenleftbigg x;h 4⎬parenrightbigg =φ(x)+hp+2ap+2(x)+O⎬parenleftbig hp+3⎬parenrightbig , show that y∗∗∗⎬parenleftbigg x;h 8⎬parenrightbigg :=2p+2y∗∗(x;h/8)−y∗∗(x;h/4) 2p+2−1 h a sar a t eo fc o n v e r g e n c eo fO( hp+3). (c)The results of using Euler’s method (with h=1 ,1/2, 1/4, 1/8) to approximate the solution to the initial value problem y/prime=y, y(0) = 1 atx= 1 are given in Table 1.2, page 26. For Euler’s method, the extrapolation procedure applies with p= 1. Use the results in Table 1.2 to find an approximation toe=y(1) by computing y∗∗∗(1;1/8). [Hint: Compute y∗(1;1/2),y∗(1;1/4), and y∗(1;1/8); then compute y∗∗(1;1/4) andy∗∗(1;1/8).] (d)Table 1.2 also contains Euler’s approximation for y(1) when h=1/16. Use this additional information to compute the next step in the extrapolation procedure; that is, compute y∗∗∗∗(1;1/16). 12 Copyright c/circlecopyrt2012 Pearson Education, Inc. Publishing as Addison-Wesley.Group Projects for Chapter 5 Effects of Hunting on Predator–Prey Systems AsdiscussedinSection5.3(pp.257–259), cyclicvariationsinthepopulationofpredators and their prey have been studied using the Volterra-Lotka predator–prey model dx dt=Ax−Bxy, (0.4) dy dt=−Cy+Dxy, (0.5) whereA,B,C,a n dDare positive constants, x(t) is the population of prey at time t,a n dy(t) is the population of predators. It can be shown that such a system has a periodic solution. That is, there exists some constant Tsuch that x(t)=x(t+T) andy(t)=y(t+T) for all t. The periodic or cyclic variation in the population has been observed in various systems such as sharks–food fish, lynx–rabbits, and ladybird beetles–cottony cushion scale. Because of this periodic behavior, it is useful to consider the average population xandydefined by x:=1 Tt⎬integraldisplay 0x(t)dt,y:=1 Tt⎬integraldisplay 0y(t)dt. (a)Show that x=C/Dandy=A/B. [Hint: Use equation (0.4) and the fact that x(0) =x(T) to show that T⎬integraldisplay 0[A−By(t)]dt=T⎬integraldisplay 0x/prime(t) x(t)dt=0. (b)Todeterminetheeffectofindiscriminatehuntingonthepopulation, assumehunting reducestherateofchangeinapopulationbyaconstanttimesthepopulation. Then the predator–prey system satisfies the new set of equations dx dt=Ax−Bxy−εx=(A−ε)x−Bxy, (0.6) dy dt=−Cy+Dxy−δy=−(C+δ)y+Dxy, (0.7) whereεandδare positive constants with ε<A. What effect does this have on the average population of prey? On the average population of predators? Copyright c/circlecopyrt2012 Pearson Education, Inc. Publishing as Addison-Wesley. 13(c)Assume the hunting was done selectively, as in shooting only rabbits (or shooting only lynx). Then we have ε>0a n dδ=0( o rε=0a n d δ>0) in (0.6)–(0.7). What effect does this have on the average populations of predator and prey? (d)In a rural county, foxes prey mainly on rabbits but occasionally include a chicken in their diet. The farmers decide to put a stop to the chicken killing by hunting the foxes. What do you predict will happen? What will happen to the farmers’ gardens? Limit Cycles In the study of triode vacuum tubes, one encounters the van der Pol equation2 y/prime/prime−μ⎬parenleftbig 1−y2⎬parenrightbig y/prime+y=0, where the constant μis regarded as a parameter. In Section 4.8, we used the mass-spring oscillator analogy to argue that the non-zero solutions to the van der Pol equation with μ= 1shouldapproachaperiodiclimitcycle. Thesameargumentappliesforanypositive value ofμ. (a)Recast the van der Pol equation as a system in normal form and use software to plot some typical trajectories for μ=0.1, 1, and 10. Re-scale the plots if necessary until you can discern the limit cycle trajectory; find trajectories that spiral in, and ones that spiral out, to the limit cycle. (b)Now letμ=−0.1,−1, and−10. Try to predict the nature of the solutions using the mass-spring analogy. Then use the software to check your predictions. Arethere limit cycles? Do the neighboring trajectories spiral into, or spiral out from, the limit cycles? (c)Repeat parts (a) and (b) for the Rayleigh equation y /prime/prime−μ⎬bracketleftBig 1−(y/prime)2⎬bracketrightBig y/prime+y=0. 2Historical Footnote : Experimental research by E. V. Appleton and B. van der Pol in 1921 on the oscillation of an electrical circuit containing a triode generator (vacuum tube) led to the non-linear equation now called 0 van der Pol’s equation . Methods of solution were developed by van der Pol in 1926–1927. Mary L. Cartwright continued research into non-linear oscillation theory and together with J. E. Little- wood obtained existence results for forced oscillations in non-linear systems in 1945. 14 Copyright c/circlecopyrt2012 Pearson Education, Inc. Publishing as Addison-Wesley.Group Project for Chapter 13 David Stapleton, University of Central Oklahoma Satellite Altitude Stability In this problem, we determine the orientation at which a satellite in a circular orbit of radiusrcan maintain a relatively constant facing with respect to a spherical primary (e.g., a planet) of mass M. The torque of gravity on the asymmetric satellite maintains the orientation. Suppose ( x,y,z)a n d(x,y,z) refer to coordinates in two systems that have a common origin at the satellite’s center of mass. Fix the xyz-axes in the satellite as principal axes; then let the z-axis point toward the primary and let the x-axis point in the direction of the satellite’s velocity. The xyz-axes may be rotated to coincide with the xyz-axes by a rotation φabout the x-axis (roll), followed by a rotation θabout the resulting y-axis (pitch), and a rotation ψabout the final z-axis (yaw). Euler’s equations from physics (withhightermsomitted3toobtainapproximatesolutionsvalidnear( φ,θ,ψ)=( 0,0,0)) show that the equations for the rotational motion due to gravity acting on the satellite are Ixφ/prime/prime=−4ω2 0(Iz−Iy)φ−ω0(Iy−Iz−Ix)ψ/prime Iyθ/prime/prime=−3ω2 0(Ix−Iz)θ Izψ/prime/prime=−4ω2 0(Iy−Ix)ψ+ω0(Iy−Iz−Ix)φ/prime, whereω0=⎬radicalbig (GM)/r3is the angular frequency of the orbit and the positive constants Ix,Iy,Izare the moments of inertia of the satellite about the x,y,a n dz-axes. (a)Find constants c1,...,c 5such that these equations can be written as two systems d dt⎡ ⎢⎢⎢⎢⎣φ ψφ /prime θ/prime⎤ ⎥⎥⎥⎥⎦=⎡ ⎢⎢⎢⎢⎣0010 0001c 100c2 0c3c40⎤ ⎥⎥⎥⎥⎦⎡ ⎢⎢⎢⎢⎣φ ψφ /prime ψ/prime⎤ ⎥⎥⎥⎥⎦ and d dt⎬bracketleftBigg θ θ/prime⎬bracketrightBigg =⎬bracketleftBigg 01 c50⎬bracketrightBigg⎬bracketleftBigg θ θ/prime⎬bracketrightBigg . 3The derivation of these equations is found in Attitude Stabilization and Control of Earth Satellites ,b y O. H. Gerlach, Space Science Reviews, #4 (1965), 541–566. Copyright c/circlecopyrt2012 Pearson Education, Inc. Publishing as Addison-Wesley. 15(b)Show that the origin is asymptotically stable for the first system in (a) if (c2c4+c3+c1)2−4c1c3>0, c1c3>0, c2c4+c3+c1>0 and hence deduce that Iy>Ix>Izyields an asymptotically stable origin. Are there other conditions on the moments of inertia by which the origin is stable? (c)Show that, for the asymptotically stable configuration in (b), the second system in (a) becomes a harmonic oscillator problem, and find the frequency of oscillation in terms of Ix,Iy,Iz,a n dω0. Phobos maintains Iy>Ix>Izin its orientation with respect to Mars, and has angular frequency of orbit ω0=0.82 rad/hr. If (Ix−Iz)/Iy=0.23, show that the period of the libration for Phobos (the period with which the side of Phobos facing Mars shakes back and forth) is about 9 hours. 16 Copyright c/circlecopyrt2012 Pearson Education, Inc. Publishing as Addison-Wesley.CHAPTER 1: Introduction EXERCISES 1.1: Background 2.This equation involves only ordinary derivatives of xwith respect to t, and the highest derivative has the second order. Thus it is an ordinary differential equation of the second order with independent variable tand dependent variable x. It is linear because x,dx/dt,a n dd2x/dt2appear in additive combination (even with constant coefficients) of their first powers. 4.This equation is an ODE because it contains no partial derivatives. Since the highest order derivative is dy/dx, the equation is a first order equation. This same term also shows us that the independent variable is xand the dependent variable is y. This equation is nonlinear because of the yin the denominator of [ y(2−3x)]/[x(1−3y)]. 6.This equation is an ordinary first order differential equation with independent variable xand dependent variable y. It is nonlinear because of the term containing the square of dy/dx. 8.This equation is an ODE because it contains only ordinary derivatives. The term dp/dt is the highest order derivative and thus shows us that this is a first order equation. This term also shows us that the independent variable is tand the dependent variable is p. This equation is nonlinear since in the term kp(P−p)=kPp−kp2the dependent variablepis squared (compare with equation (7)). 10.This equation contains only ordinary derivatives of ywith respect to x. Hence, it is an ordinary differential equation of the second order (because the highest order derivative isd2y/dx2) with independent variable xand dependent variable y. This equation is of the form (7) and, therefore, is linear. 12.ODE of the second order with the independent variable xand the dependent variable y, nonlinear. Copyright c/circlecopyrt2012 Pearson Education, Inc. Publishing as Addison-Wesley. 17Chapter 1 14.The velocity at time tis the rate of change of the position function x(t), i.e.,x/prime.T h u s , dx dt=kx4, wherekis the proportionality constant. 16.The equation is dA dt=kA2, wherekis the proportionality constant. EXERCISES 1.2: Solutions and Initial Value Problems 2. (a)Differentiating φ(x) yieldsφ/prime(x)=2x. Substitution φandφ/primeforyandy/primeinto the given equation, xy/prime=2y,g i v e sx(2x)=2(x2), which is an identity on ( −∞,∞). Thus,φ(x) is an explicit solution on ( −∞,∞). (b)We compute dφ dx=d dx(ex−x)=ex−1. Functions φ(x)a n dφ/prime(x) are defined for all real numbers and dφ dx+φ(x)2=(ex−1)+(ex−x)2 =(ex−1)+⎬parenleftbig e2x−2xex+x2⎬parenrightbig =e2x+(1−2x)ex+x2−1, which is identically equal to the right-hand side of the given equation. Thus, φ(x) is an explicit solution on ( −∞,∞). (c)Note that the function φ(x)=x2−x−1is not defined at x= 0. Differentiating φ(x) twice yields dφ dx=d dx⎬parenleftbig x2−x−1⎬parenrightbig =2x−(−1)x−2=2x+x−2; d2φ dx2=d dx⎬parenleftbiggdφ dx⎬parenrightbigg =d dx⎬parenleftbig 2x+x−2⎬parenrightbig =2+(−2)x−3=2⎬parenleftbig 1−x−3⎬parenrightbig . Therefore, x2d2φ dx2=x2·2⎬parenleftbig 1−x−3⎬parenrightbig =2⎬parenleftbig x2−x−1⎬parenrightbig =2φ(x), andφ(x)isanexplicitsolutiontothedifferentialequation x2y/prime/prime=2yonanyinterval not containing the point x= 0, in particular, on (0 ,∞). 18 Copyright c/circlecopyrt2012 Pearson Education, Inc. Publishing as Addison-Wesley.