VisualDSolve collage


Visualizing Differential Equations with Mathematica

by Antonín Slavík, Stan Wagon, and Dan Schwalbe

The VisualDSolve package was written by Dan Schwalbe and Stan Wagon in 1994 and was originally accompanied by a printed book (by Springer-Verlag) containing documentation and many examples of differential equations modeling suitable for a university course. The package provided a wide variety of tools for visualizing solutions of ordinary differential equations, including commands to generate isoclines, vector fields, flow fields, shaded phase-plane images, and Poincaré sections.

We have recently updated the package and examples to be compatible with the latest version of Mathematica, using the new features such as dynamic demonstrations, the RegionPlot function, parallel computation, and more. The package includes improved documentation in the form of an electronic book with numerous new examples and one completely new chapter with some unusual applications (see the Table of Contents below).

VisualDSolve can be purchased in the Wolfram Research bookstore.

Free sample chapters:

Antonín Slavík
Charles University
Prague, Czech Republic
Stan Wagon
Macalester College
St. Paul, MN 55105

Table of Contents

The VisualDSolve Manual

  1. VisualDSolve
    1. Overview
    2. Loading the Package
    3. Basic Usage
    4. Options
    5. Setting and Seeing the Initial Values
    6. Style and Accuracy Control
    7. Symbolic Solutions
    8. Direction Fields
    9. Isoclines
    10. Inflection Curves
    11. Controlling the Numerical Method
    12. Using the Output
    13. A Comprehensive VisualDSolve Demonstration
  2. Auxiliary Functions
    1. Overview
    2. FreehandAttempt
    3. PhaseLine
    4. ResidualPlot
    5. GetPts
    6. ToSystem
    7. ColorParametricPlot
    8. FlowParametricPlot
    9. ShowTable
  3. SystemSolutionPlot
    1. Overview
    2. Basic Usage
    3. Stylish Plots
    4. Using the Output
  4. PhasePlot
    1. Overview
    2. Basic Usage
    3. Controlling the Style of the Orbits
    4. Direction Fields, Flow Fields, and Streamlines
    5. FlowParametricPlot
    6. Controlling the t-Domain
    7. Varying a Parameter
    8. Nullcline Plots
    9. The Real Orbit Graph in Three Dimensions
    10. Systems of Three or More Equations
    11. Poincaré Sections
    12. A Phase Plane Demo
  5. SecondOrderPlot
    1. Overview
    2. A Single Second-Order Equation
    3. The Ups and Downs of a Helium Balloon
    4. Second-Order Systems

VisualDSolve and Differential Equations Modeling

  1. Differential Equations and Mathematica
    1. Rules of Solving and DSolving Equations
    2. Changing Politics: x Moves to the Right
    3. NDSolve
    4. Delay Differential Equations
    5. Expanding Dimensional Horizons
    6. Boundary Value Problems
  2. Some Parachute Experiments
    1. VisualDSolve
    2. Modeling a Parachutist
    3. Parachute to the Rescue
    4. Infinite Jerk Strikes Again: Kills Parachutist
  3. Linear Systems
    1. A Comprehensive View of the Two-Dimensional Case
    2. A Physical Application: Springs
    3. A Four-Dimensional Example
  4. Logistic Models of Population Growth
    1. One Population
    2. Two Populations
  5. Hamiltonian Systems
    1. An Example
    2. An Ideal Pendulum
    3. Higher Dimensions
  6. A Devilish Equation
    1. Skepticism Rewarded
    2. What's Going On
  7. Lead Flow in the Human Body
    1. The Model
    2. Getting the Lead Out
  8. Making a Discus Fly
    1. The Model
    2. Drag and Lift
    3. Implementing the Equations
    4. The Best Throwing Angles
  9. A Double Pendulum
    1. The Basic Pendulum
    2. Shoulder to Elbow
    3. Linearization
    4. Bringing the Pendulum to Life
    5. Chaos, and What Happens on the Way
    6. Synchronized Swinging
  10. The Duffing Equation
    1. A Stable Example
    2. The General Duffing Equation
    3. Flying Duffing Circles
    4. Forced Attraction
    5. What To Do?
    6. Strange Attraction
  11. The Tetrapods of Wada
    1. A Damped, Forced Pendulum
    2. Using the Poincaré Map
    3. Surprising Periodicity
    4. The Tetrapod
    5. The Curve-Drawing Algorithm
  12. I Tossed a Book into the Air...
    1. Euler's Equations
    2. Ellipsoids and the Conservation Laws
    3. Spinning a Book
    4. Rolling the Poinsot Ellipsoid
  13. Miscellany
    1. Fly to the Moon
    2. The Invisible Rabbit
    3. Bike Tracks
    4. Square Wheels
    5. Sharks and Fish