Mathematica

VisualDSolve

Visualizing Differential Equations with Mathematica

(Available, August 1996)

Authors
Dan Schwalbe
Department of Mathematics and Computer Science
Macalester College


Stan Wagon
Department of Mathematics and Computer Science
Macalester College

Overview
This book is a guide to VisualDSolve, a comprehensive Mathematica package that provides a wide variety of tools for the visualization of solutions to differential equations. While part of what the package does is to implement standard ideas, such as the generation of the graphs of solutions and the orbits corresponding to solutions of systems, it also includes some new ideas in the visualization of differential equations. Primary among these is the use of shaded gray regions in the phase plane, where the shading is according to the four possible directions of the underlying vector field. Another unusual idea is the use of curvy fish shapes to represent the flow of a vector field.

The material in VisualDSolve covers many of the topics generally emphasized in a first course in differential equations. Many professors are starting to emphasize numerical computation of solutions and examination of many models in the teaching of differential equations; for such courses, this book can be profitably used as a supplement, or even a textbook substitute. Students of differential equations should find the sections on modeling complex phenomena especially useful.

Copyright 1996. 256 pages/Softcover/Includes DOS-formatted diskette
ISBN 0-387-94721-3

System Requirements

The DOS-formatted diskette accompanying this book contains the Mathematica package VisualDSolve.m as well as supplemental lab notebooks as described in the book. The 3.5" diskette can be read by IBM-compatible and Macintosh computers as well as most Unix systems. Diskette files are also available online via the TELOS Web site. Version 2.0 (or later) of Mathematica is recommended for maximum use of the disk.

MathReader is a freely available program that will allow you to read (and print) any Mathematica notebook on most computer platforms.

Contents

Part 1: The VisualDSolve Manual

Chapter 1: VisualDSolve
VisualDSolve plots solutions to a single first-order ordinary differential equation. The user can specify any number of initial conditions, and options are provided to control the style of the solution curves. Important enhancements include the addition of direction fields, isocline lines, and shaded isocline regions. The method of solution can be symbolic or numeric.
Chapter 2: Auxiliary Functions
This chapter discusses some auxiliary functions that can be used with VisualDSolve. FreehandSolution sets up a situation where a student can be asked to sketch a curve on the basis of the direction field, and then compare the guess to the real solution. PhaseLine draws an image of a 1-dimensional phase line for a single autonomous differential equation. ResidualPlot produces a graph of the result when a potential solution is substituted into a differential equation.
Chapter 3: SystemSolutionPlot
SystemSolutionPlot plots the graphs of the solutions to a system of first-order differential equations. It is often true that, for a system involving equations for dx/dt and dy/dt, a phase plot (x vs. y) is more informative than individual plots of x and y as functions of t. Yet the individual plots can also be important. They show more clearly how the variables change as a function of t. SystemSolutionPlot allows the user to see the solutions either combined on a single plot or in separate plots.
Chapter 4: PhasePlot
PhasePlot plots orbits for systems of ordinary differential equations. There are several complications, since such orbits can be high-dimensional objects and can be viewed in either two or three dimensions. As usual, there are lots of options to enhance the images one can produce. The most important ones are the use of arrows or fish to represent the flow, the display of the nullcline curves and shaded nullcline regions to enhance our understanding of the flow, and the computation and classification of the equilibrium points.
Chapter 5: SecondOrderPlot
This chapter discusses the use of SecondOrderPlot to study the solutions to one or more second-order equations. This is really the culmination of the whole package since this function makes use of many of the other comprehensive functions such as PhasePlot and SystemSolutionPlot, and also because this class of equation encompasses so many fascinating examples.

Part 2: VisualDSolve and Differential Equations Modeling

Chapter 6: Differential Equations and Mathematica
The VisualDSolve package tries to make it easy to generate images of solutions to differential equations. It is nevertheless important to have an understanding of Mathematica's built-in functions for differential equations, and this chapter presents a sequence of commands and exercises to introduce those functions to students and instructors.
Chapter 7: Some Parachute Experiments
This chapter presents a lab that makes use of VisualDSolve to model a real-world problem. This lab serves both as a general introduction to the VisualDSolve package, as well as a specific introduction to some subtle aspects of a realistic problem about a parachutist.
Chapter 8: Logistic Models of Population Growth
Differential equations used to model population growth are considered in this chapter.
Chapter 9: Linear Systems
Autonomous linear systems are essentially the only large class of differential equations for which a fairly complete theory is available to solve the systems in general. They are important not only because they can be solved, but also because many applications can be closely modelled with linear systems.
Chapter 10: Hamiltonian Systems
Hamiltonian systems are special systems of differential equations that arise often in physical models. The existence of a Hamiltonian function can be very helpful in understanding the behavior of solutions to the system.
Chapter 11: A Devilish Equation
Sometimes equations that look simple cause tremendous difficulties for numerical algorithms. This chapter contains an in-depth discussion of one such equation. Users of software must always be aware of the need to look critically at the output in the hope of catching those cases, not at all rare, where numerical instabilities cause the algorithms to return incorrect images.
Chapter 12: Lead Flow in the Human Body
We discuss here a typical compartment model, in which a chemical's effect (lead) on a human is analyzed by isolating the effects on different compartments (blood, bones, tissues).
Chapter 13: Making a Discus Fly
Discus throwers have long known that a properly thrown discus goes farther against a headwind than with a following wind. The point is that a discus does in fact fly and, under certain wind conditions, the amount of lift it generates can be substantial. Our goal here is to set up a plausible model and see how large the difference in the length of a throw might be. An extra twist is the need to figure out what it means to throw a discus properly in various wind conditions.
Chapter 14: Double Pendulum
A double pendulum leads to a fairly complicated differential equation. Here we present the equations, show how to generate solutions in a variety of forms, and show how to create a movie that shows the actual pendulum swinging.
Chapter 15: The Duffing Equation
The Duffing equation is a much studied differential equation arising from a forced system (typically an oscillator subject to a periodic force). Duffing orbits exhibit a fascinating variety of behaviors typical of chaos. In this chapter we show some interesting orbits, and discuss ways to recognize and deal with sensitive situations.
Chapter 16: The Tetrapods of Wada
This chapter deals with the fairly simple differential equation governing the motion of a damped pendulum subject to a periodic external force. It seems as if, no matter the initial condition, it will eventually settle into oscillatory behavior. But how many times will it rotate before falling into the terminal loop? This question is amazingly complex: the set of initial conditions leading to oscillation after a fixed number of rotations forms a remarkable region of the plane known as a Lake of Wada. In this chapter we show how to generate a striking image of such lakes. This material is useful for the inherent beauty in the ultimate images, but also because it is a good example of the meaning and significance of the Poincaré map.
Chapter 17: Rigid Bodies
In this chapter we explore the rotational motion of a rigid body. In particular we wish to investigate the apparent instability when an object such a book is flipped in the air with the cover facing up. (View movie of spinning book.)
Appendix 1: Usage Messages
Appendix 2: Troubleshooting
References
Index

About the Authors
Dan Schwalbe is Assistant Professsor of Mathematics and Computer Science at Macalester College, St. Paul, Minnesota. Dan has been working with computer algebra systems for many years especially in the area of teaching differential equations. He is co-author of the Maple Flight Manual (1992, Brooks/Cole Publishing Company).

Stan Wagon is Professor of Mathematics and Computer Science at Macalester College, St. Paul, Minnesota. Stan is a well-known mathematical expositor, having won the MAA's Lester R. Ford prize for mathematical exposition in 1987. In addition, he is the author of the acclaimed Mathematica in Action (1991, W. H. Freeman).

View VisualDSolve home pageat Macalester College.

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