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
We solve (it's separable) giving
This is linear which we integrate giving
noting that I have absorbed some constants into .