Conditions in undetermined coefficient method

Question

I just knew about the undetermined coefficient method for solving 2nd ODE: $$y''(t) + p(t)y'(t) + q(t) = g(t)$$ and I was told that this method can only be applied when g(t) is a certain form- quasi-polynomial, I believed. So I want to ask why it only works with quasi-polynomial, and if possible what are the example and counter-example.