Chapter 8 - Linear Differential Equations of Order n - 8.3 The Method of Undetermined Coefficients: Annihilators - Problems - Page 526: 39

$y_p=xA_0e^{2x}$

We can notice that: $F(x)=7e^{2x}$ Obtain: $(D-2)(D-2)(D-3)y_p(x)= 7e^{2x} $ Therefore, the general solution for the given differential equation is: $y(x)=c_1e^{3x}+c_2 e^{2x}+xA_0e^{2x}$ The trial solution is $y_p=xA_0e^{2x}$.