Answer
See below
Work Step by Step
The general solution for the given differential equation is: $y(x)=c_1e^x+c_2 e^{-2x}+A_0e^{3x}$
The trial solution for $y_p= A_0e^{3x}$ can be computed as by plugging back into the given differential equation.
So, we have: $(D-1)(D+2)y_p(x)=5e^{3x}\\ (D^2+D-2)(A_0e^{3x})=5e^{3x}\\ 10A_0e^{3x}=5e^{3x}$
On comparing co-efficients, we get: $A_0=\frac{1}{2}$
Therefore, the general solution for the given differential equation is: $y(x)=c_1e^x+c_2e^{-2x}+\frac{1}{2}e^{3x}$
