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continued fromÌýlast page...)

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

You may note that bothÌýxÌý²¹²Ô»åÌýyÌýappear in aÌýlinearÌýform, and that they haveÌýconstant 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.ÌýThese 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 areÌýelimination, which also involves the use of linear operators, and Laplace transforms.) ButÌýMathematicaÌýhas it's own way of finding analytic solutions for itself.

Several laboratories ago you first askedÌýMathematicaÌýfor the syntax of itsÌýDSolveÌýcommand. 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.ÌýDSolveÌýshould 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 toÌýMathematicaÌýnow and give it a try.

Now let'sÌýgo and seeÌýwhat you should have gotten.