$$u'' + 4u= 8x^2 + 13e^{3x} + 16\cos2x.$$
So to find the general solution I first have to find the homogeneous and particular solutions and then general solution will be $u= u_h + u_p$.
So characteristic equation is $\lambda^2 + 4 = 0 \implies \lambda = 2i, -2i$.
So homogeneous solution is $u_h = Ae^{2ix}+ Be^{-2ix} = \cos2x + D\sin2x$.
But then I got stuck on the particular solution. Usually I would try different things to see what works but I'm unsure what to try here. Any ideas? And how do I know what to look for when trying out particular solutions?
Hint: Let $$u_p= Ax^2 + Be^{3x} + Cx\cos2x + Dx\sin2x$$