Chapter 8 - Linear Differential Equations of Order n - 8.3 The Method of Undetermined Coefficients: Annihilators - Problems - Page 525: 3

$F(x)=G(x)+3 H(x)$ Therefore, $(D^2+1)(D-2)^2$ is the annihilator of $F(x)$.

We are given that $D^2+1$ is the annihilator of $G(x)=\sin x$ Also, we have: $(D-2)^2$ is the annihilator of $H(x)=xe^{2x}$This implies that the $(D-2)^2)$ is the annihilator of $3 H(x)=3xe^{2x}$ So, $F(x)=G(x)+3 H(x)$ Therefore, $(D^2+1)(D-2)^2$ is the annihilator of $F(x)$.