Second order ODE's, proof of solution method

When solving second order ODE's, we always assume that all solutions will be proportional to e^rt.

How hard is it to prove that this has to be the case?

If there's a simple proof, please don't just state it. In stead give me a hint of where to begin, so I can try to do it myself.

If the proof is long and complicated, I don't need to know all the details (because I wouldn't understand them anyway). But it still would be nice to get a brief outline of what it involves.

Re: Second order ODE's, proof of solution method

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 .