Solving forced oscillation with non-sinusoidal periodic forcing and damping

[atomicpedals]

I've got a fairly basic question of mathematical strategy, what technique is used to solve the problem of a forced periodical (but not sinusoidal) oscillation with damping?

Do I simply find the general solution to the differential equations of motion?

 

[atomicpedals]

Or what about non-periodic oscillations?