Answer
See below
Work Step by Step
We can notice that: $F(x)=e^{x}\cos x -3e^{2x}$
Obtain: $(D^2-2D+2)^3(D-2)^2(D+4)y_p(x)=e^{x}\cos x -3e^{2x}$
Therefore, the general solution for the given differential equation is: $y(x)=c_1e^x\cos x+c_2e^{2x}+A_0\cos xe^x+A_1\sin x e^x+A_2e^{2x}$
The trial solution is $y_p=A_0\cos xe^x+A_1\sin x e^x+A_2e^{2x}$.
