Answer
See below
Work Step by Step
The general solution for the given differential equation is: $y(x)=c_1e^x+c_2 e^{-x}+A_0e^{2x}+A_1xe^{2x}$
The trial solution for $y_p=A_0e^{2x}+A_1xe^{2x}$ can be computed as by plugging back into the given differential equation.
So, we have: $y''-y=9xe^{2x}\\(D^2-1)y_p(x)=9xe^{2x}\\ 3A_0e^{2x} +4A_1e^{2x}+3A_1xe^{2x}=9xe^{2x}$
On comparing coefficients, we get: $A_0=-4,A_1=3$
Therefore, the general solution for the given differential equation is: $y(x)=c_1e^x+c_2e^{-x}-4e^{2x}+3xe^{2x}$
Given $y(0)=0\\y'(0)=7$
Substitute $c_1+c_2=4\\
c_1-c_2=5$
Then we have $c_1=8\\
c_2=-4$
Hence, $y=8e^x-4e^{-x}-4e^{2x}+3xe^{2x}$
