Question

Are all solutions of differential equations of the form Ae^(-st)?
Why do we assume it in this form?

Answer 1

Nope not all, only a simple set of simple differential equations where it works out. I think this is only ordinary linear differential equations with constant coefficients. It's effectively cause the derivatives with exponentials turns the differential equation into an algebra equation with polynomials. This is actually part of a more general approach to solving differential equations with power series / Laplace transforms and looking at linear transformations, which is also why you'll see the Wronskian floating around.

For instance, here's a differential equation I just made up:
\[xy'=2y\]
It has
\[y=Cx^2\]
as a solution.