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

.

or

.

We solve (it's separable) giving

Now

or

This is linear which we integrate giving

noting that I have absorbed some constants into .