Here's one way but I don't know if it's a proof or not. Consider the ODE
where and are constant. I'm assuming that you meant constant coefficients otherwise the solutions are not usually in the form . Now we will factor this equation using operators. We can re-write as
If we let
then (2) becomes
We solve (it's separable) giving
This is linear which we integrate giving
noting that I have absorbed some constants into .