Do we absolutely need laplace transforms to solve $y'' + ky = F_0 \sin(\sqrt k t)$? Can't we use a different method?

Question

I just saw someone solve $y'' + ky = F_0 \sin(\sqrt k t)$ using Laplace transforms and it was a real slog because it involved the unwieldy transforms of $t\sin(\sqrt k t)$ and $t\cos(\sqrt k t)$

($k, w,$ and $F_0$ are constants)

Is there a reason to do it this way instead of using a different method usable for second order linear DEs with constant coefficients? (For example finding the complementary solution and the nonhomogenous solution)?