A Course in Ordinary Differential Equations

The first contemporary textbook on ordinary differential equations (ODEs) to include instructions on MATLAB®, Mathematica®, and Maple™, A Course in Ordinary Differential Equations focuses on applications and methods of analytical and numerical solutions, emphasizing approaches used in the typical engineering, physics, or mathematics student's field of study. Stressing applications wherever possible, the authors have written this text with the applied math, engineer, or science major in mind. It includes a number of modern topics that are not commonly found in a traditional sophomore-level text. For example, Chapter 2 covers direction fields, phase line techniques, and the Runge-Kutta method; another chapter discusses linear algebraic topics, such as transformations and eigenvalues. Chapter 6 considers linear and nonlinear systems of equations from a dynamical systems viewpoint and uses the linear algebra insights from the previous chapter; it also includes modern applications like epidemiological models. With sufficient problems at the end of each chapter, even the pure math major will be fully challenged. Although traditional in its coverage of basic topics of ODEs, A Course in Ordinary Differential Equations is one of the first texts to provide relevant computer code and instruction in MATLAB, Mathematica, and Maple that will prepare students for further study in their fields.

Table of Contents:
TRADITIONAL FIRST ORDER DIFFERENTIAL EQUATIONS Some Basic Terminology Separable Differential Equations Some Physical Problems arising as Separable Equations Exact Equations Linear Equations GEOMETRICAL & NUMERICAL METHODS FOR FIRST ORDER EQUATIONS Direction Fields - the Geometry of Differential Equations Existence and Uniqueness for First Order Equations First Order Autonomous Equations - Geometrical Insight Population Modeling: An Application of Autonomous Equations Numerical Approximation with the Euler Method Numerical Approximation with the Runge-Kutta Method An Introduction to Autonomous Second Order Equations ELEMENTS OF HIGHER ORDER LINEAR EQUATIONS Some Terminology Essential Topics from Linear Algebra Reduction of Order - The Case n=2 Operator Notation Numerical Considerations for nth Order Equations TECHNIQUES OF HIGHER ORDER LINEAR EQUATIONS Homogeneous Equations with Constant Coefficients A Mass on a Spring Cauchy-Euler (Equidimensional) Equation Nonhomogeneous Equations The Method of Undetermined Coefficients via Tables The Method of Undetermined Coefficients via the Annihilator Method Variation of Parameters FUNDAMENTALS OF SYSTEMS OF DIFFERENTIAL EQUATIONS Systems of Two Equations - Motivational Examples Useful Terminology Linear Transformations and the Fundamental Subspaces Eigenvalues and Eigenvectors Matrix Exponentials TECHNIQUES OF SYSTEMS OF DIFFERENTIAL EQUATIONS A General Method, Part I: Solving Systems with Real, Distinct Eigenvalues A General Method, Part II: Solving Systems with Repeated Real or Complex Eigenvalues Solving Linear Homogeneous and Nonhomogeneous Systems of Equations Nonlinear Equations and Phase Plane Analysis Epidemiological Models LAPLACE TRANSFORMS Fundamentals of the Laplace Transform Properties of the Laplace Transforms Step Functions, Translated Functions, and Periodic Functions The Inverse Laplace Transform Laplace Transform Solution of Linear Differential Equations Solving Linear Systems using Laplace Transforms The Convolution SERIES METHODS Power Series Representations of Functions The Power Series Method Ordinary and Singular Points The Method of Frobenius Bessel Functions Appendix A: An Introduction to MATLAB, Maple, and Mathematica Appendix B: Graphing Factored Polynomials Appendix C: Selected Topics from Linear Algebra Appendix D: Answers to Selected Exercises All chapters have additional problems and each chapter has its own project(s).