\begin{definition}[Definition:Linear Differential Operator]
A '''linear differential operator''' is a differential operator $\mathscr L$ with the property that:
:$\map {\mathscr L} {\alpha \phi_1 + \beta \phi_2} = \alpha \map {\mathscr L} {\phi_1} + \beta \map {\mathscr L} {\phi_2}$
Thus if $\phi_1$ and $\phi_2$ are solutions to the differential equation $\map {\mathscr L} {\phi_i} = 0$, then so is any linear combination of $\phi_1$ and $\phi_2$.
\end{definition}