Ok, we have . Now we can solve this by, say, undetermined coefficients and obtain
But what happens when we try Laplace Transforms? It is easy to see that the Laplace transform of the equation yields which yields
So, how do we take the inverse Laplace transform of ?
Here is one way: we recognize where .
So, we might try integrating: .
(no cheating with a calculator! 🙂 )
In calculus II, we do: .
Then is transformed into (plus a constant, of course).
We now use to obtain .
Fair enough. But now we have to convert back to . We use to obtain
So converts to . Now we use the fact that as goes to infinity, has to go to zero; this means .
So what is the inverse Laplace transform of ?
Clearly, gets inverse transformed to , so the inverse transform for this part of is .
But what about the other part? so which implies that so and so the inverse Laplace transform for this part of is and the result follows.
Put another way: but since we want when and so .