solution to nonhomogeneous ODE with periodic driving force

so i found out about this after meddling with some physics and wasted about 5 sheets of paper.

if you have a 2nd order non-homogeneous linear differential equation with constant coefficient and periodic driving force like



then the particular solution to this differential equation will always be in the form of



regardless of whether it's a spring with periodic driving force, or LRC circuit. just make sure you get the ODE right.

ah, this saves so much time deriving that shit.