A linear system forced with a sinusoidal forcing function has a steady state sinusoidal reaponse. So for the particular integral take trial solution:

Now plug this into the differential equation and on the left hand side collect together all the terms with and . Equate the coefficient of the first of these to and of the second to and you will have a pair of simultaneous linear equations for and in terms of and . Solve them and you are done.

(there is an implicit assumption that is not a resonance of the system, if it is a bit more case is required)

CB