## Factorising Linear Operators

Linear operators obey their own algebra, which is in some ways similar to ordinary algebra. We can write some operators as products of operators.
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{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}

\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}