Math 118A - Differential Equations

Computing Projects

To complete the projects, you need to have access to Mathematica or a comparable computer algebra system. If you are enrolled in this class, you will be given access to the Math Department's CAMLAB, a new facility with 16 Silicon Graphics O2 machines as well as several Linux PC and PowerMAC workstations for use by faculty and students. The CAMLAB is located in PS 390.

Project 1 (with Homework Set 3):

Acquaint yourself with Mathematica. You will find the following information useful:

Unix - A Survival Guide
Using Mathematica
Revisit Problem 3 of Homework Assignment 1. Let Mathematica solve the equation, and generate the graphs of a few solutions.
You may find it useful to look at the example notebook corresponding to Chapter 1, Section 1.2, of ``Differential Equations with Mathematica''.
Revisit Problems 3 and 4a on Homework Assignment 2. Repeat the calculations with Mathematica, and graph the vector fields (for different values of lambda) as well as some phase curves. Can Mathematica find exact solutions to these equations?
You may find it useful to look at the example notebook corresponding to Chapter 1, Section 1.4 (for graphing vector fields and phase curves) as well as at Chapter 8, Sections 8.1 and 8.7 (for performing the algebraic computations) of ``Differential Equations with Mathematica''.

Project 2 (with Homework Set 4):

Verhulst, Exercise 3.5. Use Mathematica as much as possible for doing the calculation, and graph the phase space for different values of c to illustrate your argument.
Use Mathematica to study the damped oscillator equation
```
  d^2 y           d y
  -----  = -y - k ---
  d t^2           d t
```
Visualize the behavior of the eigenvalues of the system as a function of k.
Solve the equation and plot the phase portrait near the origin for each of the three distinct cases (a pair of complex conjugate eigenvalues, one real eigenvalue, two distinct real eigenvalues).
Do not use the Mathematica functions DSolve or MatrixExponential. Instead, carry out the matrix computations (change of basis!) explicitly. You may use the functions Eigenvalues, Eigenvectors, and JordanDecomposition---consult the online documentation for details.

Example Solution (in Mathematica notebook format).

Last modified: 1998/02/04
Marcel Oliver (oliver@math.uci.edu)
The Math 118 Home Page can be found at http://www.math.uci.edu/~moliver/math118/math118.html.