 ## Solving Systems

### Solving the Preliminary Example

(continued from last page...)

We're now going to use Mathematica to solve the example we encountered in the introduction to this laboratory, namely:

 x' = y - x - e3t (1) y' = 3y + 2x - 2e- t (2)

You may note that both x and y appear in a linear form, and that they have constant coefficients. In practice, with very few exceptions, only systems of this form may be solved by analytical methods. Most non-linear systems must be tackled using numerical techniques. These same restrictions will apply even when we have the computer solve systems for us, i.e. much of the time it will be necessary to have the computer use numerical rather than symbolic initial value problem solvers.

We won't be going into detail here on what the analytic techniques are for solving systems of differential equations—that discussion belongs in your lecture classes. (Two common methods are elimination, which also involves the use of linear operators, and Laplace transforms.) But Mathematica has it's own way of finding analytic solutions for itself.

Several laboratories ago you first asked Mathematica for the syntax of its DSolve command. Rather than make you repeat this, let me quote from the older laboratory:

?DSolve Notice that the second part of the description refers to a "list of differential equations". This is just another name for a system of differential equations. It looks like it won't be necessary for us to learn a new command for solving systems. DSolve should work just fine.

Based on the above, the syntax we need to solve our example system should be the following:

DSolve[{x'[t]== y[t]- x[t]- E^(3t),
y'[t]==3 y[t]+2 x[t]-2 E^(-t)},
{x[t],y[t]},t]

Jump to Mathematica now and give it a try.

Now let's go and see what you should have gotten. If you're lost, impatient, want an overview of this laboratory assignment, or maybe even all three, you can click on the compass button on the left to go to the table of contents for this laboratory assignment.

ODE Laboratories: A Sabbatical Project by Christopher A. Barker

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e-mail:
cbarker@deltacollege.edu