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I am quite confused about ordinary differential equations and the number of solutions they have. In particular, it seems that an nth order homogeneous differential equation has n solutions, not more or less. I cannot figure out why this would be so.

I have read the following two posts: here and here. The first didn't really seem to have any conclusive answers and the second was quite technical for me to understand. Perhaps there is an explanation in easy terms?

My current 'understanding' so far is as as follows:

The general solution to any homogeneous linear ODE is a linear combination of all possible solutions. In the case of an ODE of form $ay''+by'+cy=0$ where there are two equal roots to the auxillary equation, i.e. $b^2-4ac=0$, then I know the extra solution is $xe^{\lambda x}$ (although I have no idea why someone ever realised that this would work. Is there an underlying reasoning motivating this choice?)

Now if I accept this form and plug it in, I can see that indeed $xe^{\lambda x}$ is a solution iff $b^2-4ac=0$. Therefore in the case of two equal roots of the auxillary equation, we necessarily have that two of the solutions are $e^{\lambda x}$ and $xe^{\lambda x}$ so the general solution is their linear combination. And if the two roots of the auxillary equation are not equal, $xe^{\lambda x}$ is not a solution.

Of course I have not shown that there are only two solutions (could there be more?), nor have I shown that there have to be at least n solutions for an nth order ODE. I have just demonstrated that for a second order linear homogeneous ODE, $xe^{\lambda x}$ is a solution if there is only one root of the auxillary equatiuon, and it isn't otherwise.

I think my dilemma boils down to the following questions which I would be grateful if someone could answer, or indicate where to look for further information:

a)Need an nth order homogeneous ODE have n solutions (can it ever have more or less)?

b)Need an nth order linear homogeneous ODE have n solutions (can it ever have more or less)?

c)Are there any 'special cases'?

Thank you in advance.

EDIT: I have just thought a bit about 'function spaces' (i'm not sure if this is the right term. I am not too familiar with this but from my vague understanding, I think that n liearly independent solutions will span an n dimensional function space, so if it is the case that there must be n linearly independent solutions then this implies that the solution space of the ODE must span n dimensions? I am not sure why this is the case though, or what differential equations have to do with dimensions...

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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

  1. Every tuple of initial conditions $\bigl(y(0), y'(0), y''(0), \dots y^{(n-1)}(0)\bigr)$ corresponds to a solution (existence), and
  2. 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.

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To the end of the question, you have worked your way towards a more realistic picture, the initial premise is formulated unfortunately, since every differential equation has an infinity of solutions.

Look at the problem from a different perspective, the one of the existence and uniqueness theorem. It states, more or less explicitly, that there is a one-on-one relation between solutions and initial conditions. Since the initial conditions are from an $n$-dimensional space, the solution manifold inherits this dimension, trivially via the map to the values at the initial point.

See also the theory around the flow of a differential equation.

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