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.