This is an introductory textbook for students of science, mathematics, and engineering that features modeling and graphical visualization as central themes. Differential systems and numerical methods are introduced early and students are encouraged to use numerical solvers from the start. Our goal is to present this material in a way that's clear and understandable to students at all levels, that motivates them to ask ``why'' and that communicates to them our enthusiasm and excitement for the study of ODEs.

While we adopt the modern view of differential systems as evolving dynamical systems, we retain the topics and objectives of a traditional course. We introduce modern topics such as sensitivity, long-term behavior, bifurcation, and chaos, but we also present the solution formulas and theory expected in a first course.

Back to top

Dynamical Systems Approach. Throughout the text we adopt a dynamical systems approach that models natural processes that evolve in time. We treat the basic questions of existence, uniqueness, long-term behavior and sensitivity to the data as recurring themes.

Mathematical Modeling. Every picture tells a story: building a model is like drawing a picture of the system, and interpreting the solution of the model equations is like telling a story. There are a great number of models in the text from which to choose. Some sections are completely devoted to a single model, but for the most part models comprise only a portion of a section, and the text allows flexibility in the treatment of modeling.

Graphical Visualization Emphasized. The solutions of an ordinary differential equation are functions whose graphs are curves. These curves may be computer-generated and provide compelling visual evidence of mathematical deductions and a clear understanding of complicated solution formulas. Every graph in this text is accompanied by the data necessary to reproduce it. The text and the hundreds of graphs of solutions emphasize this visual connection with the theory.

Numerical Solvers Introduced Early. With the ready availability of excellent and inexpensive numerical solvers, it makes a great deal of sense to introduce a numerical solution approach very early so that students can begin to examine the geometry of solutions and the way solutions change when the elements

of a differential equation are perturbed. The introduction of computers into the course leads to heightened interest in understanding dynamical systems. The basic properties of dynamical systems serve as a valuable tool for interpreting visual displays of solutions of differential equations.

Systems Introduced Early. From the start, simple systems of differential equations are treated matter-of-factly in the modeling process, because it is natural to do so. This does not present a problem since computer solvers can handle a system of first-order differential equations as easily as a single differential equation.

Appendices. Appendix A contains proofs of the mathematical underpinnings of first-order differential equations. Appendix B contains useful background material.

Problem Sets. Problems are the heart of this book. Most sections contain some problems that require the use of a numerical solver (they are marked with a computer icon). Many sections contain open-ended projects appropriate for a team of students (marked with a handshake icon). Answers to problems with underscored numbers are given at the end of the book. Solutions to problems marked with the web icon (www) appear in this website.

Back to top

1.First-Order Differential Equations and Models
2.Initial Value Problems and Their Approximate Solutions
3.Second-Order Differential Equations
4. Applications of Second-Order Differential Equations
5. Systems of Differential Equations
6. The Laplace Transform
7. Linear Systems of Differential Equations
8. Stability
9. Cycles, Byfurcations, and Chaos
10. Fourier Series and Separation of Variables
11. Series Solutions: Bessel Functions, Legendre Polynomials
A. Basic Theory of Initial Value Problems
B. Background Information
Answers to Selected Problems
Index

Back to top