Ordinary Differential Equations: Applications, Models, and Computing(Textbooks in Mathematics)

In the traditional curriculum, students rarely study nonlinear differential equations and nonlinear systems due to the difficulty or impossibility of computing explicit solutions manually. Although the theory associated with nonlinear systems is advanced, generating a numerical solution with a computer and interpreting that solution are fairly elementary. Bringing the computer into the classroom, Ordinary Differential Equations: Applications, Models, and Computing emphasizes the use of computer software in teaching differential equations. Providing an even balance between theory, computer solution, and application, the text discusses the theorems and applications of the first-order initial value problem, including learning theory models, population growth models, epidemic models, and chemical reactions. It then examines the theory for n-th order linear differential equations and the Laplace transform and its properties, before addressing several linear differential equations with constant coefficients that arise in physical and electrical systems. The author also presents systems of first-order differential equations as well as linear systems with constant coefficients that arise in physical systems, such as coupled spring-mass systems, pendulum systems, the path of an electron, and mixture problems. The final chapter introduces techniques for determining the behavior of solutions to systems of first-order differential equations without first finding the solutions. Designed to be independent of any particular software package, the book includes a CD-ROM with the software used to generate the solutions and graphs for the examples. The appendices contain complete instructions for running the software. A solutions manual is available for qualifying instructors.

Table of Contents:
Introduction Historical Prologue Definitions and Terminology Solutions and Problems A Nobel Prize Winning Application The Initial Value Problem y′ = f (x, y); y(c) =d Direction Fields Fundamental Theorems Solution of Simple First-Order Differential Equations Numerical Solution Applications of the Initial Value Problem y′ = f (x, y); y(c) =d Calculus Revisited Learning Theory Models Population Models Simple Epidemic Models Falling Bodies Mixture Problems Curves of Pursuit Chemical Reactions N-th Order Linear Differential Equations Basic Theory Roots of Polynomials Homogeneous Linear Equations with Constant Coefficients Nonhomogeneous Linear Equations with Constant Coefficients Initial Value Problems The Laplace Transform Method The Laplace Transform and Its Properties Using the Laplace Transform and Its Inverse to Solve Initial Value Problems Convolution and the Laplace Transform The Unit Function and Time-Delay Functions Impulse Functions Applications of Linear Differential Equations with Constant Coefficients Second-Order Differential Equations Higher Order Differential Equations Systems of First-Order Differential Equations Linear Systems of First-Order Differential Equations Matrices and Vectors Eigenvalues and Eigenvectors Linear Systems with Constant Coefficients Applications of Linear Systems with Constant Coefficients Coupled Spring-Mass Systems Pendulum Systems The Path of an Electron Mixture Problems Applications of Systems of Equations Richardson’s Arms Race Model Phase-Plane Portraits Modified Richardson’s Arms Race Models Lanchester’s Combat Models Models for Interacting Species Epidemics Pendulums Duffing’s Equation Van der Pol’s Equation Mixture Problems The Restricted Three-Body Problem Appendix A: CSODE User’s Guide Appendix B: PORTRAIT User’s Guide Appendix C: Laplace Transforms Answers to Selected Exercises References Index

About the Author :
Charles E. Roberts, Jr. is a professor in the Department of Mathematics and Computer Science at Indiana State University.

Review :
Roberts gives a clear, detailed introduction to ordinary differential equations for students who have completed the full calculus sequence. … the book’s exercises and examples … are independent of any particular software—a very nice feature. Roberts provides a good balance between theoretical and applied material. … The work is very readable and offers instructors much material to work with in their courses. Summing Up: Recommended. —CHOICE, December 2010