 ## Solving Systems

(continued from last page...)

### Solving the Preliminary Example

We're going to use Mathematica to help us verify some solutions to the example we encountered in the introduction to this laboratory, namely:

 x′ = y - x - e3t (1) y′ = 3y + 2x - 2e- t (2)

We said that a solution was:

• x(t) = e- t
• y(t) = e3t

To verify that they really work as solutions, we simply need to do a little substituting. We need to substitute these values of x(t), and y(t) into both equations, and determine whether we get true statements. Let's start with equation (1):

 x′ = y - x - e3t (1)

and let's first address the left hand side. To do the check we need to substitute the proposed value of x(t) into x(t) itself, and then differentiate with respect to t. The following command does this in a single step:

D[x[t]/.x[t]->E^(-t),t]

Let's now launch Mathematica, and give the command a try. You can switch to Mathematica by clicking on the button at left. This will open up a fresh notebook for you. Remember that it will take it a while to start up! Don't forget to come back here when you're done! See you in a few minutes.

Now let's go 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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