Answer
$A_0e^{x}+A_1xe^x$
Work Step by Step
We can notice that: $F(x)=4xe^{x}$
Obtain: $(D+1)(D^2+1)y_p(x)= 4xe^{x} $
Therefore, the general solution for the given differential equation is: $y(x)=c_1e^{x}+c_2 e^{x}+A_0e^{x}+A_1xe^x$
The trial solution is $y_p=A_0e^{x}+A_1xe^x$.
