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Solution Manual for Fundamentals of Differential Equations, 9th Edition

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     ’ S S OLUTIONS M ANUAL F UNDAMENTALS OF D IFFERENTIAL E QUATIONS N INTH E DITION AND F UNDAMENTALS OF D IFFERENTIAL E QUATIONS AND B OUNDARY V ALUE P ROBLEMS S EVENTH E DITION Edward B. Saff Vanderbilt University Arthur David Snider University of South Florida

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    Contents Notes to the 1 Supplements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Computer Labs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Group Projects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Technical Writing Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Student Presentations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Homework Assignments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Syllabus Suggestions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Numerical, Graphical, and Qualitative Methods . . . . . . . . . . . . . . . . . . 3 Engineering/Physics Applications . . . . . . . . . . . . . . . . . . . . . . . . . 5 Biology/Ecology Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 Economics Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 Supplemental Group Projects 10 Detailed Solutions & Answers to Even-Numbered Problems 23 CHAPTER 1 Introduction 23 Exercises 1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 Exercises 1.2 Solutions and Initial Value Problems . . . . . . . . . . . . . . . 24 Exercises 1.3 Direction Fields . . . . . . . . . . . . . . . . . . . . . . . . . . 28 Exercises 1.4 The Approximation Method of Euler . . . . . . . . . . . . . . . 31 Review Problems Answers . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 CHAPTER 2 First Order Differential Equations 43 Exercises 2.2 Separable Equations . . . . . . . . . . . . . . . . . . . . . . . . 43 Exercises 2.3 Linear Equations . . . . . . . . . . . . . . . . . . . . . . . . . . 51 Exercises 2.4 Exact Equations . . . . . . . . . . . . . . . . . . . . . . . . . . 58 Exercises 2.5 Special Integrating Factors . . . . . . . . . . . . . . . . . . . . 66 Exercises 2.6 Substitutions and Transformations . . . . . . . . . . . . . . . . 72 Review Problems Answers . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 i

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

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

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

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    Notes to the 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: Contains complete, worked-out solutions to most odd-numbered exercises, 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 instructors in integrating computer algebra systems into their courses. Available in the Pearson Resource Center at www.pearsonhighered.com/irc . MATLAB Manual ISBN: 0321977238/9780321977236 MAPLE Manual ISBN: 0321977149/9780321977144 MATHEMATICA Manual ISBN: 0321977750/9780321977755 Computer Labs Projects Although the projects that appear at the end of the chapters in the text can be worked out by the conscientious student working alone, making them group projects may add a social element that encourages discussion and interactions that simulate a professional work place atmosphere. Group sizes of 3 or 4 seem to be optimal. Moreover, requiring that each individual 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 can be scheduled and help to improve the communication skills of the students. 1

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    2 Notes to the 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 The technical writing exercises at the end of most chapters invite students to make documented 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 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 the first few technical writing assignments with comments and have the students redo 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 solution to 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, the Projects), 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. 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 enliven their 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.

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    Notes to the 3 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 on 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 of solving differential equations include: Section 1.3: Direction Fields Section 1.4: The Approximation Method of Euler Project A for Chapter 1: Picard’s Method Project B for Chapter 1: The Phase Line Project D for Chapter 1: Taylor Series Method 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).

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    4 Notes to the 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 F for Chapter 3: Stability of Numerical Methods Project G for Chapter 3: Period Doubling and Chaos 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 Δ u = f on a Rectangle Project D for Chapter 11: Shooting Method Project E for Chapter 11: Finite-Difference Method for Boundary Value Problems Section 12.8: Neurons and the FitzHugh-Nagumo Equations , which uses direction fields to establish the onset of action potentials on axons. Project C for Chapter 12: Computing Phase Plane Diagrams Project D for Chapter 12: Ecosystem of Planet GLIA-2

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    Notes to the 5 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 Procedure for Equations Appendix G: Software for Analyzing Differential Equations The instructor who wishes to emphasize numerical methods should also note that the text contains an extensive chapter on 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 extensive discussion in Chapter 12. Further material dealing with engineering/physics applications include: Project A for Chapter 2: Oil Spill in a Canal 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 .

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    6 Notes to the Project 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-entry Project C for Chapter 4: Simple Pendulum Project H for Chapter 4: Gravity Train 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 G for Chapter 5: Phase-Locked Loops 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

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    Notes to the 7 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. 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 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 C for Chapter 1: The Phase Plane , which discusses the logistic population model and bifurcation diagrams for population control. Problem 40 in Exercises 2.3, which discusses the Hodgkins-Huxley model for axon ac- tivity Project A for Chapter 2: Oil Spill in a Canal .

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    8 Notes to the 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 . Project B for Chapter 3: Aquaculture , which deals with a model of raising and harvesting 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. 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: 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 Equations Project D for Chapter 12: Ecosystem on Planet GLIA-2 Project E for Chapter 12: Spread of Staph Infections in Hospitals – Part II . Economics Applications Project C for Chapter 1: Applications to Economics Project H for Chapter 2: Utility Functions and Risk Aversion Project E for Chapter 3: Market Equilibrium: Stability and Time Paths

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    Notes to the 9 The basic content of the remainder of this instructor’s manual consists of supplemental projects, 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 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 projects. E. B. Saff A. D. Snider Edward.B.Saff@Vanderbilt.edu snider@.usf.edu

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    Projects for Chapter 3 Delay Differential Equations In our discussion of mixing problems in Section 3.2, we encountered the initial value problem x ′ ( t ) = 6 − 3 500 x ( t − t 0 ) , (0.1) x ( t ) = 0 for x ∈ [ − t 0 , 0] , where t 0 is 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 − t 0 ) 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 ′ = ax ( t − b ) , (0.2) where a , b are constants, has a solution of the form x ( t ) = Ce st for any constant C , provided s satisfies 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 that x ( t ) = f ( t ) for − b ≤ t ≤ 0. For 0 ≤ t ≤ b , equation (0.2) becomes x ′ ( t ) = ax ( t − b ) = af ( t − b ) , and so x ( t ) = t ∫ 0 af ( ν − b ) dν + x (0) . Now that we know x ( t ) on [0 , b ], we can repeat this procedure to obtain x ( t ) = t ∫ b ax ( ν − b ) dν + x ( b ) for b ≤ x ≤ 2 b . This process can be continued indefinitely. 1 See, 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

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