Chapter 8 - Linear Differential Equations of Order n - 8.3 The Method of Undetermined Coefficients: Annihilators - Problems - Page 525: 10

$F(x)= G(x)-2H(x)+3 J(x)-K(x)$ Therefore, $(D-4)^2 (D^2-8D+41) D^2 (D^2 +4D+5)^3$ is the annihilator of $F(x)$.

We found that $D^2-8D+41$ is the annihilator of $-2 H(x)=-2 e^{4x} \sin 5x$ and $D^2$ is the annihilator of $3 J(x) = 3x$ Next, $(D^2+4D+5)^3$ is the annihilator of $-k(x)=-x^{2} e^{-2x} \cos x$ So, $F(x)= G(x)-2H(x)+3 J(x)-K(x)$ Therefore, $(D-4)^2 (D^2-8D+41) D^2 (D^2 +4D+5)^3$ is the annihilator of $F(x)$.