Second Order ODE
Oh, okay. Your problem is in your guess for the particular solution. You should note that it's the same as the homogeneous solution, so of course when you plug it into the differential equation, you get 0.
The problem is that the forcing function looks like the homogeneous solution. That is, both contain the term ##\sin \omega t##. When this happens, you need to multiply your guess for the particular solution by t to get solutions to the differential equation independent of the homogeneous solution. So try ##x_p(t) = Ct\cos \omega t + Dt\sin \omega t##.
