How to Modify Y(t) for Nonhomogenous 2nd Order DE with e^-t and cos(2t) Terms?

[jesuslovesu]

Homework Statement

Well I've got another one that totally sucks.
y'' + 2y' + 5y = 4e^{-t}cos(2t)

Homework Equations

The Attempt at a Solution

I tried Y(t) = Ae^{-t}cos(2t) + Be^{-t}sin(2t) but that unfortunately yielded 0 = 4e^{-t} cos(2t)

So my question is how does one modify Y(t) in this type of situation? The only thing I can think of is something like Y(t) = Ae^{-t}t^2cos(2t) + Be^{-t}tsin(2t) but that seems rather painful

 

[Dick]

Try,
<br /> Y(t) = Ae^{-t}tcos(2t) + Be^{-t}tsin(2t)<br />
which is what you wrote down but I changed a t^2 to a t. Yeah, it's kind of painful, but it will work. Without the t's it just the homogeneous solution. You knew that would give you zero, right?