Chapter 8 - Linear Differential Equations of Order n - 8.3 The Method of Undetermined Coefficients: Annihilators - Problems - Page 526: 42

See below

We can notice that: $F(x)=4x+9xe^{2x} $ Obtain: $(D^2+4)(D-2)^3y_p(x)= 4x+9xe^{2x} $ Therefore, the general solution for the given differential equation is: $y(x)=c_1x+c_2x e^{2x} +A_0x+A_1x+A_2e^{2x}+A_3e^{2x}$ The trial solution is $y_p=A_0x+A_1x+A_2e^{2x}+A_3e^{2x}$.