Chapter 8 - Linear Differential Equations of Order n - 8.3 The Method of Undetermined Coefficients: Annihilators - Problems - Page 525: 13

$D(D^2+4)$ is the annihilator of $F(x)$

We have: $F(x)=\dfrac{1+ \cos 2x}{2}$ Re-write as: $F(x)=\dfrac{1}{2}+\dfrac{\cos 2x}{2}$ So, $D$ is the annihilator of $\dfrac{1}{2}$ and $D^2+4$ is the annihilator of $\dfrac{\cos 2x}{2}$. Therefore, $D(D^2+4)$ is the annihilator of $F(x)$.