I'm having trouble solving ordinary differential equations (DE's), because I don't understand the underlying theory, nor how to apply it.
For example, suppose I am solving such a DE. My techniques allow me to find the general solution on an interval $(a,b),$ and also on an interval $(b,c).$ Under what circumstances can I, by gluing pairs of solutions, obtain the general solution on the larger interval $(a,c)$ ? In particular, under what circumstances can I conclude that all solutions on $(a,c)$ can be obtained in this way?
Is there a book (article, lecture series etc.) that explains these kinds of things?
Here's what I'm looking for in a book.
- Precisely stated definitions and theorems.
- Examples of how to apply them in order to solve problems rigorously. (By "rigorous solution to a problem," let us mean a solution that not only gets the right answer, but also constitutes a proof that the answer is correct.)
- An author with a keen sense for the difference between an argument that can easily be made rigorous, versus an argument that sounds convincing but which upon closer inspection is full of holes.
Here's what I don't (currently) need.
- Formal proofs of the theorems.
- PDE's (yet!).