Trying to get some geometric intuition on differential equations

[farleyknight]

I want to preface this by saying that these questions are not to find an exact answer, just to build intuition. If you find them ill-posed or incorrect, it would be most helpful if you could show me a "better way" of looking at it.


So, I'm trying to gather a geometric viewpoint of differential equations. From what I understand, the solutions of a differential equation form a manifold. Is that correct? I have yet to study manifolds from a serious point of view, but there is enough literature out there to suggest that this might be the case.

Secondly, and I'm assuming that the previous statement is true, for the case of exact equations: Are allowed to use integrating factors because they are diffeomorphisms on the underlying manifold? I'm guessing that because we can stretch the solution a bit, one way or another, that we still end up with the same phase space.

 

[gtoguy97]

The last question I have, and again assuming that the previous statement is correct, is about the general solution of an equation. How do we determine the general solution of an equation from a geometrical perspective? Does it have something to do with understanding the underlying manifold structure and then finding a transformation of variables that simplifies the equation? Any help here would be greatly appreciated.