Chapter 8 - Linear Differential Equations of Order n - 8.3 The Method of Undetermined Coefficients: Annihilators - Problems - Page 526: 45

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We can notice that: $F(x)=2e^{3x}+e^{2x}\sin x$ Obtain: $D(D^2-9)(D^2-4D+5)y_p(x)=2e^{3x}+e^{2x}\sin x$ Therefore, the general solution for the given differential equation is: $y(x)=c_1e^{3x}+c_2e^{2x}\sin x+A_0e^{3x}+A_1\sin x e^{2x}+A_2\cos x e^{2x}$ The trial solution is $y_p=A_0e^{3x}+A_1\sin x e^{2x}+A_2\cos x e^{2x}$.