Solution Manual For Fundamentals Of Differential Equations, 8th Edition

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ContentsNotes to the Instructor1Supplements. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1Computer Labs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2Group Projects. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2Technical Writing Exercises. . . . . . . . . . . . . . . . . . . . . . . . . . . . .2Student Presentations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3Homework Assignments. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3Syllabus Suggestions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4Numerical, Graphical, and Qualitative Methods. . . . . . . . . . . . . . . . . .4Engineering/Physics Applications. . . . . . . . . . . . . . . . . . . . . . . . .6Biology/Ecology Applications. . . . . . . . . . . . . . . . . . . . . . . . . . . .8Supplemental Group Projects10Detailed Solutions & Answers to Even-Numbered Problems17CHAPTER 1Introduction17Exercises 1.1Detailed Solutions. . . . . . . . . . . . . . . . . . . . . . . . .17Exercises 1.2Detailed Solutions. . . . . . . . . . . . . . . . . . . . . . . . .18Exercises 1.3Detailed Solutions. . . . . . . . . . . . . . . . . . . . . . . . .23Exercises 1.4Detailed Solutions. . . . . . . . . . . . . . . . . . . . . . . . .25Tables. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .29Figures. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .30CHAPTER 2First Order Differential Equations35Exercises 2.2Detailed Solutions. . . . . . . . . . . . . . . . . . . . . . . . .35Exercises 2.3Detailed Solutions. . . . . . . . . . . . . . . . . . . . . . . . .43Exercises 2.4Detailed Solutions. . . . . . . . . . . . . . . . . . . . . . . . .51Exercises 2.5Detailed Solutions. . . . . . . . . . . . . . . . . . . . . . . . .59Exercises 2.6Detailed Solutions. . . . . . . . . . . . . . . . . . . . . . . . .65Review ProblemsAnswers. . . . . . . . . . . . . . . . . . . . . . . . . . . .74Tables. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .75Figures. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .76©i

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CHAPTER 3Mathematical Models and Numerical MethodsInvolving First Order Equations77Exercises 3.2Detailed Solutions. . . . . . . . . . . . . . . . . . . . . . . . .77Exercises 3.3Detailed Solutions. . . . . . . . . . . . . . . . . . . . . . . . .85Exercises 3.4Detailed Solutions. . . . . . . . . . . . . . . . . . . . . . . . .91Exercises 3.5Answers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .100Exercises 3.6Answers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .101Exercises 3.7Answers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .101Tables. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .103Figures. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .104CHAPTER 4Linear Second Order Equations105Exercises 4.1Detailed Solutions. . . . . . . . . . . . . . . . . . . . . . . . .105Exercises 4.2Detailed Solutions. . . . . . . . . . . . . . . . . . . . . . . . .107Exercises 4.3Detailed Solutions. . . . . . . . . . . . . . . . . . . . . . . . .115Exercises 4.4Detailed Solutions. . . . . . . . . . . . . . . . . . . . . . . . .123Exercises 4.5Detailed Solutions. . . . . . . . . . . . . . . . . . . . . . . . .128Exercises 4.6Detailed Solutions. . . . . . . . . . . . . . . . . . . . . . . . .140Exercises 4.7Detailed Solutions. . . . . . . . . . . . . . . . . . . . . . . . .147Exercises 4.8Detailed Solutions. . . . . . . . . . . . . . . . . . . . . . . . .161Exercises 4.9Detailed Solutions. . . . . . . . . . . . . . . . . . . . . . . . .164Exercises 4.10Detailed Solutions. . . . . . . . . . . . . . . . . . . . . . . . .171Review ProblemsAnswers. . . . . . . . . . . . . . . . . . . . . . . . . . . .175Figures. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .177CHAPTER 5Introduction to Systems and Phase Plane Analysis181Exercises 5.2Answers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .181Exercises 5.3Answers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .183Exercises 5.4Answers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .184Exercises 5.5Answers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .186Exercises 5.6Answers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .186Exercises 5.7Answers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .186Exercises 5.8Answers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .187Review ProblemsAnswers. . . . . . . . . . . . . . . . . . . . . . . . . . . .188Tables. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .189Figures. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .191CHAPTER 6Theory of Higher-Order Linear Differential Equations197Exercises 6.1Answers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .197Exercises 6.2Answers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .198Exercises 6.3Answers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .198Exercises 6.4Answers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .199ii©

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Review ProblemsAnswers. . . . . . . . . . . . . . . . . . . . . . . . . . . .200CHAPTER 7Laplace Transforms201Exercises 7.2Detailed Solutions. . . . . . . . . . . . . . . . . . . . . . . . .201Exercises 7.3Detailed Solutions. . . . . . . . . . . . . . . . . . . . . . . . .205Exercises 7.4Detailed Solutions. . . . . . . . . . . . . . . . . . . . . . . . .211Exercises 7.5Detailed Solutions. . . . . . . . . . . . . . . . . . . . . . . . .219Exercises 7.6Detailed Solutions. . . . . . . . . . . . . . . . . . . . . . . . .228Exercises 7.7Detailed Solutions. . . . . . . . . . . . . . . . . . . . . . . . .242Exercises 7.8Detailed Solutions. . . . . . . . . . . . . . . . . . . . . . . . .250Exercises 7.9Detailed Solutions. . . . . . . . . . . . . . . . . . . . . . . . .255Review ProblemsAnswers. . . . . . . . . . . . . . . . . . . . . . . . . . . .265Figures. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .266CHAPTER 8Series Solutions of Differential Equations271Exercises 8.1Answers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .271Exercises 8.2Answers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .272Exercises 8.3Answers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .273Exercises 8.4Answers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .274Exercises 8.5Answers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .275Exercises 8.6Answers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .275Exercises 8.7Answers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .277Exercises 8.8Answers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .278Review ProblemsAnswers. . . . . . . . . . . . . . . . . . . . . . . . . . . .279Figures. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .280CHAPTER 9Matrix Methods for Linear Systems281Exercises 9.1Answers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .281Exercises 9.2Answers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .281Exercises 9.3Answers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .282Exercises 9.4Answers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .285Exercises 9.5Answers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .287Exercises 9.6Answers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .289Exercises 9.7Answers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .290Exercises 9.8Answers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .292Review ProblemsAnswers. . . . . . . . . . . . . . . . . . . . . . . . . . . .293Figures. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .295CHAPTER 10Partial Differential Equations297Exercises 10.2Answers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .297Exercises 10.3Answers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .297Exercises 10.4Answers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .298©iii

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Exercises 10.5Answers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .299Exercises 10.6Answers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .300Exercises 10.7Answers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .301CHAPTER 11Eigenvalue Problems and Sturm-Liouville Equations303Exercises 11.2Answers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .303Exercises 11.3Answers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .304Exercises 11.4Answers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .304Exercises 11.5Answers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .305Exercises 11.6Answers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .306Exercises 11.7Answers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .308Exercises 11.8Answers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .309Review ProblemsAnswers. . . . . . . . . . . . . . . . . . . . . . . . . . . .309CHAPTER 12Stability of Autonomous Systems311Exercises 12.2Answers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .311Exercises 12.3Answers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .311Exercises 12.4Answers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .312Exercises 12.5Answers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .312Exercises 12.6Answers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .313Exercises 12.7Answers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .313Review ProblemsAnswers. . . . . . . . . . . . . . . . . . . . . . . . . . . .314Figures. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .315CHAPTER 13Existence and Uniqueness Theory323Exercises 13.1Answers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .323Exercises 13.2Answers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .323Exercises 13.3Answers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .324Exercises 13.4Answers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .324Review ProblemsAnswers. . . . . . . . . . . . . . . . . . . . . . . . . . . .324Appendix A:Review of Integration Techniques325Exercises ADetailed Solutions. . . . . . . . . . . . . . . . . . . . . . . . . .325iv©

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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.SupplementsStudent’s Solutions Manual:By Viktor Maymeskul. Contains complete, worked-out solu-tions to most odd-numbered exercises, providing students with an excellent study tool. ISBN13: 978-0-321-74834-8; ISBN 10: 0-321-74834-4.Companion Web site:Provides additional resources for both instructors and students,including helpful links keyed to sections of the text, access to Interactive Differential Equations,suggestions for incorporating Interactive Differential Equations modules, suggested syllabi,index of applications, and study tips for students. Access:www.pearsonhighered.com/nagleInteractive Differential Equations:By Beverly West (Cornell University), Steven Strogatz(Cornell University), Jean Marie McDill (California Polytechnic State University – San LuisObispo), John Cantwell (St. Louis University), and Hubert Hohn (Massachusetts College ofArts) is a popular software directly tied to the text that focuses on helping students visualizeconcepts. Applications are drawn from engineering, physics, chemistry, and biology. Access:www.pearsonhighered.com/nagleInstructor’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 instructorsin integrating computer algebra systems into their courses. Available in the Pearson InstructorResource Center atwww.pearsonhighered.com/irc.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: 0-321-52178-1©1

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Computer LabsA 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 collegesand 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, 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 togetherwith 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 themgroupprojects 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 10).Technical Writing ExercisesThe technical writing exercises at the end of most chapters invite students to make documentedresponses to questions dealing with the concepts in the chapter. This not only gives studentsan opportunity to improve their writing skills, but it helps them organize their thoughts andbetter understand the new concepts.Moreover, many questions deal with critical thinking2©

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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 returnungradedthe first few technical writing assignments with comments and have the students redothe 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 PresentationsIt 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 excellentcourse to allow (require) a student to give a presentation on a special application (e.g., almostany topic from Chapters 3 and 5), on a new technique not covered in class (e.g., material fromSection 2.6, Projects A, B, or C in Chapter 4), or on additional theory (e.g., material fromChapter 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 substantiallyto 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 aidof video technology, and utilizing appropriate internet web sites.Homework AssignmentsWe would like to share with you an obvious, non-original, but effective method to encouragestudents 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. Wethen choose at random one problem from each assignment (typically, a total of 3) that willbe graded. (The point is that the student does not know in advance which problems will bechosen.) Full credit is given for any of the chosen problems for which there is evidence that thestudent has made an honest attempt at solving. The homework problem sets are returned tothe students at the next meeting (Mondays) with grades like 0/3, 1/3, 2/3, or 3/3 indicatingthe proportion of problems for which the student received credit. The homework grades aretallied at the end of the semester and count as one test grade. Certainly, there are variations©3

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on this theme. The point is that students are motivated to do their homework.Syllabus SuggestionsTo 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. Asa 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 MethodsThe sections and projects dealing with numerical, graphical, and qualitative techniques ofsolving differential equations include:Section 1.3:Direction FieldsSection 1.4:The Approximation Method of EulerProject A for Chapter 1:Taylor Series MethodProject B for Chapter 1:Picard’s MethodProject C for Chapter 1:The Phase LineSection 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. Theseoutlines 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 includesoutlines for the Fourth Order Runge-Kutta subroutine and algorithm with tolerance (seepp. 135–136).Project H for Chapter 3:Stability of Numerical MethodsProject I for Chapter 3:Period Doubling and Chaos4©

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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 describesthe vectorized forms of Euler’s method and the Fourth Order Runge-Kutta method, anddiscusses an application to population dynamics.Section 5.4:Introduction to the Phase Plane, which introduces the study of trajectoriesof autonomous systems, critical points, and stability.Section 5.8:Dynamical Systems, Poincar´e Maps, and Chaos, which discusses the use ofnumerical methods to approximate the Poincar`e map and how to interpret the results.Project A for Chapter 6:Computer Algebra Systems and Exponential ShiftProject D for Chapter 6:Higher-Order Difference EquationsProject A for Chapter 8:Alphabetization AlgorithmsProject D for Chapter 10:Numerical Method forΔu=fon a RectangleProject D for Chapter 11:Shooting MethodProject E for Chapter 11:Finite-Difference Method for Boundary Value ProblemsProject C for Chapter 12:Computing Phase Plane DiagramsProject D for Chapter 12:Ecosystem of Planet GLIA-2Section 13.1:Introduction: Successive ApproximationsAppendix B:Newton’s MethodAppendix C:Simpson’s RuleAppendix E:Method of Least SquaresAppendix F:Runge-Kutta Procedure for Equations©5

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The instructor who wishes to emphasize numerical methods should also note that the textcontains an extensive chapter of series solutions of differential equations (Chapter 8).Engineering/Physics ApplicationsSince Laplace transforms is a subject vital to engineering, we have included a detailed chapteron this topic – see Chapter 7. Stability is also an important subject for engineers, so we haveincluded an introduction to the subject in Section 5.4 along with an entire chapter addressingthis topic – see Chapter 12.Further material dealing with engineering/physics applicationsinclude: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 problemsand of population models.Section 3.3:Heating and Cooling off Buildings, which discusses temperature variationsin 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©

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Section 4.10:A Closer Look at Forced Mechanical Vibrations.Project B for Chapter 4:Apollo Re-entryProject C for Chapter 4:Simple PendulumSection 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 discussionsof transfer functions, the Dirac delta function, and frequency response modelling.Project B for Chapter 8,Spherically Symmetric Solutions to Schr¨odinger’s Equation forthe Hydrogen AtomProject D for Chapter 8,Buckling of a TowerProject E for Chapter 8,Aging Spring and Bessel FunctionsSection 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, theheat equation, wave equation, and Laplace’s equation.©7

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Project 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-conservativeautonomous 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 particularlyuseful, especially Section 8.8 on special functions.Biology/Ecology ApplicationsProject C for Chapter 1:The Phase Plane, which discusses the logistic population modeland 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 problemsand population models.Project A for Chapter 3:Dynamics of HIV Infection.Project B for Chapter 3:Aquaculture, which deals with a model of raising and harvestingcatfish.Section 5.1:Interconnected Fluid Tanks, which introduces systems of equations.Section 5.3:Solving Systems and Higher-Order Equations Numerically, which containsan application to population dynamics.8©

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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 LakesProject 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-2Project E for Chapter 12:Spread of Staph Infections in Hospitals – Part II.Project F for Chapter12:A Growth Model for Phytoplankton – Part II.The basic content of the remainder of this instructor’s manual consists of supplemental groupprojects, 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 placeelse since the text and theStudent’s Solutions Manualonly provide answers and solutions toodd-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 encouragesharing with us (the authors and users of the texts) any of your favorite group projects.E. B. SaffA. D. SniderEdward.B.Saff@Vanderbilt.edusnider@eng.usf.edu©9

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Solution Manual For Fundamentals Of Differential Equations, 8th Edition - Page 15 preview image

Group Projects for Chapter 3Delay Differential EquationsIn our discussion of mixing problems in Section 3.2, we encountered the initial valueproblemx′(t) = 6−3500x(t−t0),(0.1)x(t) = 0forx∈[−t0,0],wheret0is a positive constant. The equation in (0.1) is an example of adelay differ-ential equation.These equations differ from the usual differential equations by thepresence of the shift (t−t0) in the argument of the unknown functionx(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 equationx′=ax(t−b),(0.2)wherea,bare constants, has a solution of the formx(t) =Cestfor any constantC, providedssatisfies the transcendental equations=ae−bs.(b)A solution to (0.2) fort >0 can also be found using themethod of steps. Assumethatx(t) =f(t) for−b≤t≤0. For 0≤t≤b, equation (0.2) becomesx′(t) =ax(t−b) =af(t−b),and sox(t) =t∫0af(ν−b)dν+x(0).Now that we knowx(t) on [0, b], we can repeat this procedure to obtainx(t) =t∫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, NewYork, 1963, orOrdinary and Delay Differential Equations, by R. D. Driver, Springer–Verlag, New York, 197710©

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Solution Manual For Fundamentals Of Differential Equations, 8th Edition - Page 16 preview image

Use the method of steps to show that the solution to the initial value problemx′(t) =−x(t−1),x(t) = 1on[−1,0],is given byx(t) =n∑k=0(−1)k[t−(k−1)]kk!,forn−1≤t≤n ,wherenis a nonnegative integer.(This problem can also be solved using theLaplace transform method of Chapter 7.)(c)Use the method of steps to compute the solution to the initial value problem givenin (0.1) on the interval 0≤t≤15 fort0= 3.ExtrapolationWhen precise information about theformof the error in an approximation is known, atechnique calledextrapolationcan be used to improve the rate of convergence.Suppose the approximation method converges with rate O (hp) ash→0 (cf. Section 3.6).From theoretical considerations, assume we know, more precisely, thaty(x;h) =φ(x) +hpap(x) + O(hp+1),(0.3)wherey(x;h) is the approximation toφ(x) using step sizehandap(x) is some functionthat is independent ofh(typically, we do not know a formula forap(x), only that itexists).Our goal is to obtain approximations that converge at the faster rate thanO (hp+1).We start by replacinghbyh/2 in (0.3) to gety(x;h2)=φ(x) +hp2pap(x) + O(hp+1).If we multiply both sides by 2pand subtract equation (0.3), we find2py(x;h2)−y(x;h) = (2p−1)φ(x) + O(hp+1).Solving forφ(x) yieldsφ(x) = 2py(x;h/2)−y(x;h)2p−1+ O(hp+1).©11

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