You are here

Revolutions in Differential Equations: Exploring ODEs with Modern Technology

Michael J. Kallaher
Mathematical Association of America
Publication Date: 
Number of Pages: 
MAA Notes 50
[Reviewed by
Jan Holly
, on

A revolution is rumbling through the teaching of differential equations. Computers are beginning to inundate the classroom, and courses are changing in response to the availability of software. Although computers have been used for a long time in differential equations courses, the last few years has seen an explosion in software written specifically for such classes and textbooks focusing on graphical methods. Hence, the mainstream now follows the revolutionists.

This new MAA volume consists of articles presenting a host of ideas for making use of technology in teaching differential equations. Included are ideas for classroom examples, student exercises, and ways to structure a course. Also included are descriptions of available software and references to Web resources. The level of information varies, from textbook-like presentations of basic graphical methods, to discussions of surprising mathematical phenomena, to summaries of research-level differential equations software. The ideal audience for this volume is a teacher who is not yet familiar with the newest (e.g. the last five years') textbooks and software; however, anyone who teaches differential equations can benefit from the plethora of ideas. These ideas come from an set of experienced people, who are the articles' authors: Robert L. Borrelli & Courtney S. Coleman, William E. Boyce, Michael Branton & Margie Hale, Kevin D. Cooper & Thomas LoFaro, David O. Lomen, Valipuram S. Manoranjan, Lawrence F. Shampine & Ian Gladwell, and Beverly H. West.

The most exciting component of this volume is the array of interesting examples and exercises. Borrelli & Coleman show a number of fun examples with fancy graphics, including 3-D graphs. Lomen gives exercises that use data, having the student explore different possible forms of differential equations that might fit, for example, the velocity data obtained when, "A student shot a 120 grain hollow point bullet from a 300 Winchester Magnum rifle." One example by Manoranjan is the analysis of an equilibrium point that turns out to be a nonlinear spiral masquerading as a center. The article by Branton & Hale, like many of the articles, is presented in textbook form, explaining the concepts and including some exercises. Many of the examples in this volume are similar to those already found in modern textbooks, but there is enough additional variety here to make the volume worthwhile. Every article, regardless of specific focus, includes ideas of exercises and/or examples for use in the classroom.

Also useful are the discussions of computer resources. Various software packages are discussed, and various Web sites are referenced (some excellent but some outdated, as is to be expected). However, the volume fails to achieve its full potential in the area of software reviews; an index is lacking, so the reader who wants to learn about particular software must search through the text of individual articles in hopes of finding tidbits. Borrelli & Coleman give an overview of three packages: Interactive Differential Equations (IDE), Internet Differential Equations Activities (IDEA), and ODE Architect. Many of the other existing packages are mentioned by other authors. In addition, Shampine & Gladwell focus on numerical methods and include a summary of relevant software, and Cooper & LoFaro talk about the Internet and its possibilities.

To incorporate computers fully into a course, the structure of the course, and possibly the classroom itself, must be revamped. The topic of class structure is treated well in this volume. West, especially, gives many ideas and suggestions on how to incorporate computer exercises and projects, including helpful hints on assigning, grading, and presenting results of student work. Boyce, also, discusses class structure, advocating the "studio course" in which students have access to computers at all times in the classroom.

The revolution is worth joining, and this volume is worth having, for anyone who teaches differential equations. Although the reader may wish for a better catalogue and review of software, or a longer list of relevant books (no-one mentions as part of the revolution, for example, the excellent textbook Nonlinear Dynamics and Chaos by Steven H. Strogatz, 1994, Addison-Wesley, Reading, Mass.), the volume serves its purpose well by presenting a number of ideas for using technology in teaching differential equations.

Jan Holly ( is Clare Boothe Luce Assistant Professor of Mathematics at Colby College, where she specializes in applied mathematics and teaches differential equations.