Chapter 8 - Linear Differential Equations of Order n - 8.3 The Method of Undetermined Coefficients: Annihilators - Problems - Page 525: 4

$F(x)=G(x)+5H(x)$ Therefore, $(D-7)^4(D^2+16)$ is the annihilator of $F(x)$.

We are given that $(D-7)^4$ is the annihilator of $G(x)=x^3 e^{7x}$ Also, we have: $D^2+16$ is the annihilator of $H(x)=\cos 4x$.This implies that the $D^2+16$ is the annihilator of $5 H(x)=5 \cos 4x$ So, $F(x)=G(x)+5H(x)$ Therefore, $(D-7)^4(D^2+16)$ is the annihilator of $F(x)$.