Second order non homogeneous DE, annihilator method

I've followed Chris' tutorial, but I didn't get lucky at the end.

I must solve . Also, and .

So I consider the homogeneous equation . I wrote the characteristic polynomial and found its roots to be . Thus a solution to the homogeneous DE is . I'm not sure if I can solve for and yet, with the initial conditions. It would be the solution the homogeneous DE, but are these constants the same in the case of the non homogeneous case? I guess yes...

Anyway, I continued writing . The annihilator is .

Hence . So or (this one is a double root).

Thus a general solution to the original DE is .

I tried to solve for and this way:

.

.

.

Plugging these values into the original DE, I reach that cancels out... Obviously I made something wrong. I don't see what.