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
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
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."