Springer Verlag New York: published by TELOS, 1997, xiv+271 p

ISBN 0 387 94721 3 (paperback)

**Review by André Heck**

CAN/Amstel Institute

University of Amsterdam, The Netherlands

Email: `heck@can.nl`

The title of this book refers to the most important procedure in a *Mathematica*
package with the same name that provides a variety of tools for visualization of ordinary
differential equations. The package and the book cover many topics of a first course
in differential equations: direction fields, isoclines, phaseplot, and
Poincaré section, to name a few.
However, emphasis is on graphical investigations of differential equations and not
on methods of finding or approximating solutions symbolically. The package not only
implements standard methods for graphing solutions and orbits corresponding to
solutions of systems, but also other graphical methods for investigating ordinary
differential equations, such as grayshading of phase planes according to the four
possible directions of the underlying vector field and use of curvy fish shapes to
represent the flow of a vector field.

The book consists of two parts: the first five chapters form the guide to the
`VisualDSolve`

package and the last twelve chapters illustrate ways to use the package.
The diskette accompanying the book contains the *Mathematica* package as well as
supplemental lab notebooks as described in the book. The material on the diskette is also
available online via the TELOS web site at `www.telospub.com`

.

The first part of the book is a clearly written user's guide to the main procedures
in the package and their optional arguments.
The first chapter explains the `VisualDSolve`

command that plots solutions of a
single first-order ordinary differential equations. The user can specify any number of
initial conditions and options are provided to control the style of solution curves
and to add direction fields, isocline lines, and/or shaded isocline regions.

The second chapter discusses some auxiliary functions that can be used with `VisualDSolve`

.
`FreehandAttempt`

sets up a situation to sketch a solution curve on the basis
of a direction field and to compare it to the real solution. `PhaseLine`

draws an image of a 1-dimensional phase line for a single autonomous
differential equation and `ResidualPlot`

produces the graph of the result when a
potential solution is substituted into a differential equation.

The third chapter describes the procedure `SystemSolutionPlot`

, which plots
as its name suggests the graphs of the solutions of a systems of first-order differential
equations.

The two main routines discussed in the fourth chapter are `PhasePlot`

, which plots orbits
for systems of ordinary differential equations, and `PoincareSection`

for constructing
Poincaré sections of first-order systems of two equations or of a single second-order
equation.
Images can be enhanced via options: the
most important ones are the use of arrows or fish to represent the flow, the display of
nullline curves and shaded nullcline regions, and the computation and classification of
equilibrium points

The fifth chapter discussed the use of `SecondOrderPlot`

to study te solutions of one
or more second-order differential equations.

The first two chapters of the second part of the book
are lab notebooks, meant to be used in an interactive way by students to familiarize themselves
with the basics of *Mathematica* (in particular `DSolve`

and `NDSolve`

)
and the `VisualDSolve`

package. The other chapters are about modeling with
differential equations. Here, you will find traditional topics such as linear systems (spring
systems), nonlinear systems (logistic models of population growth), and
Hamiltonian systems (pendulum), but also nonstandard examples such as
the chaotic double pendulum, the flight of a discus, and the transfer of lead in a human body.
In separate chapters, the Duffing equation and the differential
equation governing the motion of the damped periodically driven pendulum
are used to investigate the role of initial conditions on the behavior of solutions:
chaotic behavior of solutions and Wada basins, i.e. regions of initial conditions leading
to oscillation after a fixed number of rotations, are the keywords here.
The last chapter of the book is a very nice case study on tossing a book into the air;
an example of the rotational motion of a rigid body.

In the preface, the authors state that they hope that their package and book can be profitably used as a supplement in either an introductory differential equations course or a modeling course. In the reviewer's opinion, they have certainly succeeded in this; they have provided teachers and students with easy-to-use tools and with relevant, attractive examples that invite to further explorations.