Must any nth order homogeneous ODE have n solutions?
Solution 1:
Let me restrict attention to the linear case. Then the space of solutions is a vector space, and one can ask what its dimension is. The answer, for an $n^{th}$ order homogeneous linear ODE (with constant coefficients, to be completely precise), is that it is always $n$-dimensional. This means you can find a basis of it consisting of $n$ linearly independent solutions, but there are in general many such bases. (And there are many more than $n$ solutions; if $n$ is positive there are infinitely many solutions.)
This is a consequence of the existence and uniqueness theorems for ODEs, which say that
- Every tuple of initial conditions $\bigl(y(0), y'(0), y''(0), \dots y^{(n-1)}(0)\bigr)$ corresponds to a solution (existence), and
- A solution $y$ is completely determined by its initial conditions $y(0), y'(0), y''(0), \dots y^{(n-1)}(0)$ (uniqueness).
So the reason the space of solutions is $n$-dimensional is that the space of initial conditions is $n$-dimensional.
The question of what these solutions actually look like requires a more detailed analysis and that's a bit of a separate question.