Chapter 8 - Linear Differential Equations of Order n - 8.3 The Method of Undetermined Coefficients: Annihilators - Problems - Page 525: 16

$D(D^2+64)$ is the annihilator of $F(x)$.

We have: $F(x)=\sin^2 x \cos^2 x \cos^2 2x$ Re-write as: $F(x)=(\dfrac{\sin^2(2x)}{4}) \cos^2 (2x)=\dfrac{1}{36}(1-\cos 8x)$ So, $D$ is the annihilator of $\dfrac{1}{36}$ and $D^2+64$ is the annihilator of $\dfrac{-\cos 8x}{36}$ . Therefore, $D(D^2+64)$ is the annihilator of $F(x)$.