What is the solution to a nonhomogenous second order differential equation?

[jesuslovesu]

[SOLVED] Nonhomogenous Second Order DE

Homework Statement

$$y'' - 2y' - 3y = -3te^{-t}$$
Solve

Homework Equations

The Attempt at a Solution

Well first I tried
$$Y(t) = Ate^{-t}$$
then
$$Y(t) = At^2e^{-t}$$
then
$$Y(t) = At^3e^{-t}$$
and
$$Y(t) = At^3e^{-t} + Bt^2e^{-t}$$
but I can't seem to get the correct form.

Does anyone know what form I need my Y(t) in order to solve this?

 

[jesuslovesu]

I believe I end up with a term like
$$Ae^{-t} = -3te^{-t}$$ which I don't believe I can do anything with?

In fact, in all the cases I've tried I always seem to get something like $$Ae^{-t}$$ which never cancels out so I can never equation -3te^{-t} to any of my terms

 

[jesuslovesu]

ah finally got it with $$At^2e^{-1}+Bte^{-1}$$
bah that sucked