Solving the Preliminary Example
continued from听last page...)
Your check of equation (2) should look like this:
D[y[t]/.y[t]->E^(3t),t]
3 y[t]+2 x[t]-2 E^(-t)/.{x[t]->E^(-t),y[t]->E^(3t)}
听
Clearly both sides give the same result, making the second equation true when you substitute the proposed solution into it. We have now verified that the proposed solution does indeed satisfy both differential equations in the system. We say that we have found听a听solution of the system. We can't claim that we've found the system's听general听solution, since that would be a听family听of solutions. What we've found is just a single member of that family.
Let's go听and tackle the problem more seriously, putting听Mathematica's "big guns" to work.
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