Solution Manual for Fundamentals of Differential Equations, 9th Edition

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’SSOLUTIONSMANUALFUNDAMENTALS OFDIFFERENTIALEQUATIONSNINTHEDITIONANDFUNDAMENTALS OFDIFFERENTIALEQUATIONSANDBOUNDARYVALUEPROBLEMSSEVENTHEDITIONEdward B. SaffVanderbilt UniversityArthur David SniderUniversity of South Florida

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ContentsNotes to the1Supplements. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1Computer Labs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1Group Projects. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1Technical Writing Exercises. . . . . . . . . . . . . . . . . . . . . . . . . . . . .2Student Presentations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2Homework Assignments. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2Syllabus Suggestions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3Numerical, Graphical, and Qualitative Methods. . . . . . . . . . . . . . . . . .3Engineering/Physics Applications. . . . . . . . . . . . . . . . . . . . . . . . .5Biology/Ecology Applications. . . . . . . . . . . . . . . . . . . . . . . . . . . .7Economics Applications. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .8Supplemental Group Projects10Detailed Solutions & Answers to Even-Numbered Problems23CHAPTER 1Introduction23Exercises 1.1Background. . . . . . . . . . . . . . . . . . . . . . . . . . . . .23Exercises 1.2Solutions and Initial Value Problems. . . . . . . . . . . . . . .24Exercises 1.3Direction Fields. . . . . . . . . . . . . . . . . . . . . . . . . .28Exercises 1.4The Approximation Method of Euler. . . . . . . . . . . . . . .31Review ProblemsAnswers. . . . . . . . . . . . . . . . . . . . . . . . . . . .35Tables. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .36Figures. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .38CHAPTER 2First Order Differential Equations43Exercises 2.2Separable Equations. . . . . . . . . . . . . . . . . . . . . . . .43Exercises 2.3Linear Equations. . . . . . . . . . . . . . . . . . . . . . . . . .51Exercises 2.4Exact Equations. . . . . . . . . . . . . . . . . . . . . . . . . .58Exercises 2.5Special Integrating Factors. . . . . . . . . . . . . . . . . . . .66Exercises 2.6Substitutions and Transformations. . . . . . . . . . . . . . . .72Review ProblemsAnswers. . . . . . . . . . . . . . . . . . . . . . . . . . . .80i

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iiContentsTables. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .82Figures. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .82CHAPTER 3Mathematical Models and Numerical MethodsInvolving First Order Equations85Exercises 3.2Compartmental Analysis. . . . . . . . . . . . . . . . . . . . .85Exercises 3.3Heating and Cooling of Buildings. . . . . . . . . . . . . . . . .93Exercises 3.4Newtonian Mechanics. . . . . . . . . . . . . . . . . . . . . . .99Exercises 3.5Electrical Circuits. . . . . . . . . . . . . . . . . . . . . . . . .108Exercises 3.6Improved Euler’s Method. . . . . . . . . . . . . . . . . . . . .109Exercises 3.7Higher-Order Numerical Methods: Taylor and Runge-Kutta. .109Tables. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .111Figures. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .112CHAPTER 4Linear Second Order Equations113Exercises 4.1Introduction: The Mass-Spring Oscillator. . . . . . . . . . . .113Exercises 4.2Homogeneous Linear Equations: The General Solution. . . . .115Exercises 4.3Auxiliary Equations with Complex Roots. . . . . . . . . . . .123Exercises 4.4Nonhomogeneous Equations: The Method of UndeterminedCoefficients. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .131Exercises 4.5The Superposition Principle and Undetermined CoefficientsRevisited. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .136Exercises 4.6Variation of Parameters. . . . . . . . . . . . . . . . . . . . . .148Exercises 4.7Variable-Coefficient Equations. . . . . . . . . . . . . . . . . .155Exercises 4.8Qualitative Considerations for Variable-Coefficient and NonlinearEquations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .168Exercises 4.9A Closer Look at Free Mechanical Vibrations. . . . . . . . . .171Exercises 4.10A Closer Look at Forced Mechanical Vibrations. . . . . . . .177Review ProblemsAnswers. . . . . . . . . . . . . . . . . . . . . . . . . . . .182Figures. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .184CHAPTER 5Introduction to Systems and Phase Plane Analysis189Exercises 5.2Elimination Method for Systems with Constant Coefficients. .189Exercises 5.3Solving Systems and Higher–Order Equations Numerically. . .191Exercises 5.4Introduction to the Phase Plane. . . . . . . . . . . . . . . . .192Exercises 5.5Applications to Biomathematics:Epidemic and Tumor GrowthModels. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .194Exercises 5.6Coupled Mass–Spring Systems. . . . . . . . . . . . . . . . . .194Exercises 5.7Electrical Systems. . . . . . . . . . . . . . . . . . . . . . . . .195Exercises 5.8Dynamical Systems, Poincar`e Maps, and Chaos. . . . . . . . .195Review ProblemsAnswers. . . . . . . . . . . . . . . . . . . . . . . . . . . .196Tables. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .197

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ContentsiiiFigures. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .199CHAPTER 6Theory of Higher-Order Linear Differential Equations205Exercises 6.1Basic Theory of Linear Differential Equations. . . . . . . . . .205Exercises 6.2Homogeneous Linear Equations with Constant Coefficients. . .206Exercises 6.3Undetermined Coefficients and the Annihilator Method. . . .206Exercises 6.4Method of Variation of Parameters. . . . . . . . . . . . . . . .207Review ProblemsAnswers. . . . . . . . . . . . . . . . . . . . . . . . . . . .208CHAPTER 7Laplace Transforms209Exercises 7.2Definition of the Laplace Transform. . . . . . . . . . . . . . .209Exercises 7.3Properties of the Laplace Transform. . . . . . . . . . . . . . .213Exercises 7.4Inverse Laplace Transform. . . . . . . . . . . . . . . . . . . . .219Exercises 7.5Solving Initial Value Problems. . . . . . . . . . . . . . . . . .227Exercises 7.6Transforms of Discontinuous Functions. . . . . . . . . . . . . .236Exercises 7.7Transforms of Periodic and Power Functions. . . . . . . . . . .245Exercises 7.8Convolution. . . . . . . . . . . . . . . . . . . . . . . . . . . . .250Exercises 7.9Impulses and the Dirac Delta Function. . . . . . . . . . . . . .258Exercises 7.10Solving Linear Systems with Laplace Transforms. . . . . . . .263Review ProblemsAnswers. . . . . . . . . . . . . . . . . . . . . . . . . . . .273Figures. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .274CHAPTER 8Series Solutions of Differential Equations279Exercises 8.1Introduction: The Taylor Polynomial Approximation. . . . . .279Exercises 8.2Power Series and Analytic Functions. . . . . . . . . . . . . . .280Exercises 8.3Power Series Solutions to Linear Differential Equations. . . . .281Exercises 8.4Equations with Analytic Coefficients. . . . . . . . . . . . . . .282Exercises 8.5Cauchy-Euler (Equidimensional) Equations Revisited. . . . . .283Exercises 8.6Method of Frobenius. . . . . . . . . . . . . . . . . . . . . . . .283Exercises 8.7Finding a Second Linearly Independent Solution. . . . . . . .285Exercises 8.8Special Functions. . . . . . . . . . . . . . . . . . . . . . . . . .286Review ProblemsAnswers. . . . . . . . . . . . . . . . . . . . . . . . . . . .287Figures. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .288CHAPTER 9Matrix Methods for Linear Systems289Exercises 9.1Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . .289Exercises 9.2Review 1: Linear Algebraic Equations. . . . . . . . . . . . . .289Exercises 9.3Review 2: Matrices and Vectors. . . . . . . . . . . . . . . . . .290Exercises 9.4Linear Systems in Normal Form. . . . . . . . . . . . . . . . .293Exercises 9.5Homogeneous Linear Systems with Constant Coefficients. . . .295Exercises 9.6Complex Eigenvalues. . . . . . . . . . . . . . . . . . . . . . .297Exercises 9.7Nonhomogeneous Linear Systems. . . . . . . . . . . . . . . . .298

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ivContentsExercises 9.8The Matrix Exponential Function. . . . . . . . . . . . . . . .300Review ProblemsAnswers. . . . . . . . . . . . . . . . . . . . . . . . . . . .301Figures. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .303CHAPTER 10Partial Differential Equations305Exercises 10.2Method of Separation of Variables. . . . . . . . . . . . . . . .305Exercises 10.3Fourier Series. . . . . . . . . . . . . . . . . . . . . . . . . . .305Exercises 10.4Fourier Cosine and Sine Series. . . . . . . . . . . . . . . . . .306Exercises 10.5The Heat Equation. . . . . . . . . . . . . . . . . . . . . . . .307Exercises 10.6The Wave Equation. . . . . . . . . . . . . . . . . . . . . . . .308Exercises 10.7Laplace’s Equation. . . . . . . . . . . . . . . . . . . . . . . .309CHAPTER 11Eigenvalue Problems and Sturm-Liouville Equations311Exercises 11.2Eigenvalues and Eigenfunctions. . . . . . . . . . . . . . . . .311Exercises 11.3Regular Sturm-Liouville Boundary Value Problems. . . . . .312Exercises 11.4Nonhomogeneous Boundary Value Problems and the FredholmAlternative. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .312Exercises 11.5Solution by Eigenfunction Expansion. . . . . . . . . . . . . .313Exercises 11.6Green’s Functions. . . . . . . . . . . . . . . . . . . . . . . . .314Exercises 11.7Singular Sturm-Liouville Boundary Value Problems. . . . . .316Exercises 11.8Oscillation and Comparison Theory. . . . . . . . . . . . . . .317Review ProblemsAnswers. . . . . . . . . . . . . . . . . . . . . . . . . . . .317CHAPTER 12Stability of Autonomous Systems319Exercises 12.2Linear Systems in the Plane. . . . . . . . . . . . . . . . . . .319Exercises 12.3Almost Linear Systems. . . . . . . . . . . . . . . . . . . . . .319Exercises 12.4Energy Methods. . . . . . . . . . . . . . . . . . . . . . . . .320Exercises 12.5Lyapunov’s Direct Method. . . . . . . . . . . . . . . . . . . .320Exercises 12.6Limit Cycles and Periodic Solutions. . . . . . . . . . . . . . .321Exercises 12.7Stability of Higher-Dimensional Systems. . . . . . . . . . . .321Exercises 12.8Neurons and FitzHugh-Nagumo Equations. . . . . . . . . . .322Review ProblemsAnswers. . . . . . . . . . . . . . . . . . . . . . . . . . . .325Figures. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .326CHAPTER 13Existence and Uniqueness Theory335Exercises 13.1Introduction: Successive Approximations. . . . . . . . . . . .335Exercises 13.2Picard’s Existence and Uniqueness Theorem. . . . . . . . . .335Exercises 13.3Existence of Solutions of Linear Equations. . . . . . . . . . .336Exercises 13.4Continuous Dependence of Solutions. . . . . . . . . . . . . .336Review ProblemsAnswers. . . . . . . . . . . . . . . . . . . . . . . . . . . .336Appendix A:Review of Integration Techniques337Exercises ADetailed Solutions. . . . . . . . . . . . . . . . . . . . . . . . . .337

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Notes to theOne 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:Contains complete, worked-out solutions to most odd-numberedexercises, providing students with an excellent study tool.Companion Web site:’s MAPLE/MATLAB/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 PearsonResource Center atwww.pearsonhighered.com/irc.MATLAB Manual ISBN: 0321977238/9780321977236MAPLE Manual ISBN: 0321977149/9780321977144MATHEMATICA Manual ISBN: 0321977750/9780321977755Computer LabsProjectsAlthough the projects that appear at the end of the chapters in the text can be worked outby the conscientious student working alone, making themgroupprojects may add 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 can be scheduled and help to improve the communication skills of the students.1

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2Notes to theThe 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 thinkingskills 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., materialfrom Section 2.6, the Projects), or on additional theory (e.g., material from Chapter 6 whichgeneralizes the results in Chapter 4). In addition to improving students’ communication skills,these “special” topics are long remembered by the students. Working in groups of 3 or 4 andsharing the presentation responsibilities can add substantially to the interest and quality of thepresentation. Students should also be encouraged to enliven their communication by buildingphysical models, preparing part of their lectures with the aid of video technology, and utilizingappropriate 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.

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Notes to the3We 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 variationson 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:Picard’s MethodProject B for Chapter 1:The Phase LineProject D for Chapter 1:Taylor Series MethodSection 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).

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4Notes to theSection 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 F for Chapter 3:Stability of Numerical MethodsProject G for Chapter 3:Period Doubling and ChaosSection 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 ProblemsSection 12.8:Neurons and the FitzHugh-Nagumo Equations, which uses direction fieldsto establish the onset of action potentials on axons.Project C for Chapter 12:Computing Phase Plane DiagramsProject D for Chapter 12:Ecosystem of Planet GLIA-2

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Notes to the5Section 13.1:Introduction: Successive ApproximationsAppendix B:Newton’s MethodAppendix C:Simpson’s RuleAppendix E:Method of Least SquaresAppendix F:Runge-Kutta Procedure for EquationsAppendix G:Software for Analyzing Differential EquationsThe instructor who wishes to emphasize numerical methods should also note that the textcontains an extensive chapter on 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 extensive discussion inChapter 12. Further material dealing with engineering/physics applications include:Project A for Chapter 2: Oil Spill in a CanalProject 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.

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6Notes to theProject D for Chapter 3:Aircraft Guidance in a Crosswind.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.Section 4.10:A Closer Look at Forced Mechanical Vibrations.Project B for Chapter 4:Apollo Re-entryProject C for Chapter 4:Simple PendulumProject H for Chapter 4:Gravity TrainSection 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 G for Chapter 5:Phase-Locked LoopsProject 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 Atom

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Notes to the7Project 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.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 E for Chapter 10:The Telegrapher’s Equation and the Cable EquationProject 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.Problem 40 in Exercises 2.3, which discusses the Hodgkins-Huxley model for axon ac-tivityProject A for Chapter 2:Oil Spill in a Canal.

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8Notes to theProject 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.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:The 2014-2015 Ebola Epidemic.Problem 19 in Exercises 10.5 , which involves chemical diffusion through a thin layer.Section 12.8:Neurons and the Fitz-Nagumo EquationsProject D for Chapter 12:Ecosystem on Planet GLIA-2Project E for Chapter 12:Spread of Staph Infections in Hospitals – Part II.Economics ApplicationsProject C for Chapter 1:Applications to EconomicsProject H for Chapter 2:Utility Functions and Risk AversionProject E for Chapter 3:Market Equilibrium: Stability and Time Paths

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Notes to the9The basic content of the remainder of this instructor’s manual consists of supplementalprojects, answers to the even-numbered problems, and detailed solutions to most of them.These answers are not available any place else since the text and theStudent’s SolutionsManualonly 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 encouragesharing with us (the authors and users of the texts) any of your favorite projects.E. B. SaffA. D. SniderEdward.B.Saff@Vanderbilt.edusnider@.usf.edu

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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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