Is this a property of commutative differential operators? I found this claim in a paper but the proof escapes me. I'm sure that it's simple. 
Suppose we have $\psi$, a solution to the ODE $LQ\psi=0$, where $L$ and $Q$ are commutative differential operators. Then it is possible to write $\psi=\psi_1+\psi_2$ such that $L\psi_1=0$ and $Q\psi_2=0$.
Thanks

If $L, Q$ are commutative then $LQ-QL=0$. Then $LQ \psi = LQ(\psi_{1}+\psi_{2}) = QL(\psi_{1}+\psi_{2})$. hence
\begin{eqnarray}
LQ(\psi_{1}+\psi_{2}) &=& LQ\psi_{1}+LQ\psi_{2}\\
                      &=&  L(Q\psi_{1}+Q\psi_{2}) \\ 
\end{eqnarray}
Run this through with the other appropriate relation and the assumption should be clear.
