Example: If
\[D= \frac{d}{dx}\]
then \[(D^2-3D+2)y=(D-1)(D-2)y\]
.If
\[y=xe^x\]
then \[Dy= \frac{dy}{dx}= e^x+xe^x=(1+x)e^x, \: D^2y=\frac{d^2y}{dx^2}=e^x+e^x+xe^x=(2+x)e^x\]
.\[\begin{equation} \begin{aligned} (D^2-3D+2)(xe^x) &= (2+x)e^x-3(x+1)e^x+2xe^x \\ &= e^x(2+x-3x-3+2x) \\ &= -e^x \end{aligned} \end{equation}\]
\[\begin{equation} \begin{aligned} (D-2)(D-1)(xe^x) &= (D-2)(xe^x+e^x-xe^x) \\ &= (D-2)(e^x) \\ &= e^x-2e^x \\ &= -e^x \end{aligned} \end{equation}\]