Answer
See below
Work Step by Step
The general solution for the given differential equation is: $y(x)=c_1e^{-5x}+c_2 e^{2x}+A_0xe^{2x}$
The trial solution for $y_p= A_0xe^{2x}$ can be computed as by plugging back into the given differential equation.
So, we have: $(D-2)(D+5)y_p(x)=14e^{2x}\\ (D^2+3D-10)(A_0xe^{2x})=14e^{2x}\\ 7A_0e^{2x} +3A_0xe^{2x}=14e^{2x}$
On comparing co-efficients, we get: $A_0=2$
Therefore, the general solution for the given differential equation is: $y(x)=c_1e^{-5x}+c_2 e^{2x}+2xe^{2x}$
