Solving the Preliminary Example

continued fromÌýlast page...)

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

³æ′Ìý=ÌýyÌý-ÌýxÌý-Ìýe3t (1)
²â′Ìý= 3yÌý+ 2xÌý- 2e-Ìýt (2)

We said that a solution was:

  • x(t) =Ìýe-Ìýt
  • y(t) =Ìýe3t

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

³æ′Ìý=ÌýyÌý-ÌýxÌý-Ìýe3t ÌýÌý(1)

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

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

Let's now launchÌýMathematica, and give the command a try.

Mathematica LogoYou can switch toÌýMathematicaÌýby 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'sÌýgo seeÌýwhat you should have gotten...

Compass 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.