Chapter 8 - Linear Differential Equations of Order n - 8.3 The Method of Undetermined Coefficients: Annihilators - Problems - Page 525: 2

$F(x)=2G(x)-3 H(x)$ Therefore, $(D-1)D^2=D^3-D^2$ is the annihilator of $F(x)$,

We consider the function $F(x)=2e^{x}-3x$ We are given that $D-1$ is the annihilator of $G(x)=e^{x}$ This implies that the $2G(x)=2e^x$ and so, $F(x)=2G(x)-3 H(x)$ Therefore, $(D-1)D^2=D^3-D^2$ is the annihilator of $F(x)$.