Differential Equations, Live and in Color

"A picture may be worth a thousand words, but a good animation is worth much more," says Selwyn Hollis, Professor of Mathematics at Armstrong Atlantic State University.

Hollis has made many contributions to the use of advanced technology in mathematical education. He has authored a number of software supplements to Stewart's popular calculus textbooks. His publications about Mathematica show how students can use it as a tool to explore calculus concepts and applications without getting bogged down by algebraic and computational details.

A Mathematica Companion for Differential Equations, Hollis's most recent book, is designed to supplement a typical college-level differential equations course and shows students how to use Mathematica to solve and visualize common differential equations.

The book's companion website demonstrates how animations can illustrate practical applications of calculus. Imagine, for example, poking a hole in the bottom of a plastic cup filled with water. How quickly will the water drain out?

Hollis generated a Mathematica animation that simultaneously simulates this simple experiment and plots the results. Both representations originate from the differential equations that relate the change in water level to the shape of the container and to the volume of water. By seeing the representations develop side by side, students can better understand how the equations, graph, and physical situation all relate to each other.

Watch the animation and see if you can come up with explanations for what you see. Why, for example, does the water level change more slowly at the end? You could go one step further and try the same thing with a plastic cup and a sink. Are you convinced that the differential equations and animations describe the real world accurately?

Other examples on the website include springs, pendulums, population models, circuits, decay models, numerical techniques, and chaotic systems. The examples demonstrate concepts and methods of differential calculus such as nonlinear equations, orbits, vector fields, and phase diagrams.

By using Hollis's A Mathematica Companion for Differential Equations in conjunction with his custom differential equation packages for Mathematica, students can reproduce the animations, experiment with the input parameters, and perhaps even discover powerful animations of their own. Says Hollis, who plans to keep adding new material to his website, "I'm always interested in new ideas, and with their fertile imaginations, students often provide good ones."