In most explanations, you are given something like
$ay''+by'+cy=0$
and the method starts with something like, "I can see that the exponential function would be a solution, and also adding two exponential functions together would work, so..." and so on.
What if I can't just magically see that this solution covers every possible solution? How do you find a solution for this kind of equation algebraically? I thought I had come up with a solution in an earlier question (https://math.stackexchange.com/a/200082/21050), but I didn't understand why my solution worked so that's no good.
I just learned a clever way to solve these when $c=0$, but assume that $c$ is not zero for your answer, please.
edit: I want to alter this question slightly: I can go ahead and assume that $e^{mx}$ is a solution and work from there. But what if I was braindead and couldn't make that guess? How can I get $e^{mx}$ to spit out somewhere?