Answer
See below
Work Step by Step
The general solution for the given differential equation is: $y(x)=c_1+c_2 e^{-3x}+A_0x+A_1e^x+A_2xe^x$
The trial solution for $y_p= A_0x+A_1e^x+A_2xe^x$ can be computed as by plugging back into the given differential equation.
So, we have: $D(D+3)y_p(x)=x(5+e^x)\\ (D^2+3D)(A_0x+A_1e^x+A_2xe^x)=x(5+e^x)\\ (A_1+2A_2+3A_1)e^x+(A_2+3A_2)xe^x=5x+xe^x$
On comparing coefficients, we get: $A_0=\frac{5}{3}$ and $A_1=-\frac{1}{8},A_2=\frac{1}{4}$
Therefore, the general solution for the given differential equation is: $y(x)=c_1+c_2 e^{-3x}+\frac{5}{3}x+\frac{1}{4}e^x-\frac{1}{8}xe^x$
