Solving the Preliminary Example

continued from last page...)

Your session should have gone like this:

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]

What a messy answer! A far cry from the simple pair of functions that I suggested you try to start with. But remember, this isn't just "any old solution" you have found. This is the almighty general solution. Within this solution lie all others, including the simple one we started out the lab with! (Indeed, if you look carefully over the above solution, you should notice that multiples of our original solutions of e^- t and e^3t are lurking within.)

It is worth noting here that if you were to use the elimination method for solving systems that you'll be seeing in the lecture part of your course, the solutions that Mathematica produces bear little resemblance to your answers on paper—especially at first glance. This is because Mathematica's internal algorithms don't use the elimination method, but instead use a much more mechanical procedure better suited to coding into a computer. Unfortunately, this often results in some pretty ugly looking answers. Rest assured that the results can be shown to be equivalent, after some extensive manipulation.

For much of what we'll be using systems of differential equations for in this class, it will not be necessary to find an analytic solution, and we'll be perfectly happy to make do with a numerical one. Indeed, in many cases we will have no choice, since the systems in question can't be solved analytically anyway. So let's now move on and see how Mathematica handles numerical solutions to systems of differential equations.

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.