After taking the Laplace transform, where did the come from?
Just wondering if someone could have a quick look over what I've done so far:
Assuming zero initial conditions, solve the following DE:
The Laplace Transform gives:
Rearranging and using the fact that gives us
So ?
Agreed. You could simply say that you extend the solution found by the LT method to the entire real line. The result of the inverse LT does have the unit step functions in it, and, in fact, does not satisfy the DE for negative t's. However, with the theorem you have invoked, you can extend the solution by eliminating the unit step function.