Chapter 8 - Linear Differential Equations of Order n - 8.3 The Method of Undetermined Coefficients: Annihilators - Problems - Page 525: 7

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

Since, $(D-4)^2$ is the annihilator of $G(x)= xe^{4x}$ Also, we have: $(D-4)^2$ is the annihilator of $H(x)=e^{4x}$.This implies that the $(D-4)^2$ is the annihilator of $H(x)-3G(x)=e^{4x}-3xe^{4x}$ So, $F(x)= H(x)-3 G(x)+2 J(x)$ Therefore, $D^3(D-4)^2$ is the annihilator of $F(x)$.