Chapter 8 - Linear Differential Equations of Order n - 8.3 The Method of Undetermined Coefficients: Annihilators - Problems - Page 526: 43

See below

We can notice that: $F(x)=11e^{x}-\sin 2x $ Obtain: $D^2(D-1)(D^2+4)^2y_p(x)=11e^{x}-\sin 2x $ Therefore, the general solution for the given differential equation is: $y(x)=c_1e^x+c_2\sin 2x+A_0e^{x}+A_1\sin2x+A_2\cos 2x$ The trial solution is $y_p=A_0e^{x}+A_1\sin2x+A_2\cos 2x$.