This may be a duplicate, though I have searched and not found this question answered, and it may also belong more on Mathematics Stack exchange than here -- in which case I'll transfer.
My question is: how does one prove (both intuitively and rigorously) that the solution to the SHO, being a linear combination of a sine and cosine, is the most general and unique solution?
The way it is most often solved is by simply suggesting $x(t) = \exp(\Omega t)$, then solving $\Omega = \pm ik/m$, and ending up with $x(t) = A\cos(\omega t) + B\sin(\omega t)$ with $\omega = |\Omega|$.
Suppose I am going trough this derivation with a high-school physics enthusiast, and he/she asks me "You've simply supposed $x(t)$ to be exponential, and showed that if it is, the solution is $\dots$, how do you know this is $\textit{the}$ solution?". I've done a differential equations class, and even though I passed it, I seem to have missed this crucial aspect.
EDIT
Since the posting of this question, two answers have been posted only answering the question of $\textit{how}$ the SHO should be solved. A question I did not ask.
My question has boiled down to this; how do I show that the space of solutions of the SHO (and any 2nd order ODE) is two-dimensional? This would answer my question.