Undetermined Coefficients

So far the right side was zero. But real systems get pushed: a periodic force, a steady load, a ramp. Now g(x) ≠ 0. The full answer splits in two — the natural response y_h (the homogeneous solution you already know) plus a particular response y_p that matches the forcing. Undetermined coefficients is the fastest way to find y_p: guess its SHAPE, then solve for the numbers.