Proof for ODE Existence and Uniqueness Theorems

[Pingy]

My current math professor refuses to prove any of the theorems he states in class.

He is all about practicality, and it kinda bugs me because all the Calc based courses I have taken so far have proven all the theorems stated.

Does anyone here know the proofs for The Existence and Uniqueness of Solutions Theorems for ODEs? Or at least where I could find them?

I love reading proofs because I always learn a lot about whatever is being proved. It really helps me grasp the concept and how to apply it much better than without.

[bobfish]

I've had math and science teachers like that. It's because as teachers they may have hundreds of students each year so they can't teach every student in the way they learn best. So they use the method that tends to work for most students. Since most student's are mediocre in most subjects, the students with the greatest potential get the short end of the stick.

[danreil]

Here's a sketch of the proof for a general non-linear first order ODE which you should be able to understand if you're taking a class in ODEs. http://preview.tinyurl.com/5jksnv This really is a sketch though, but it does give some idea of a proof.


Here's a much more complete proof, but unless you know some functional analysis or advanced differential equations, you might not understand everything, like the Banach fixed point theorem and the Lispschitz property, but you can just take them as given and go through the proof anyway. Btw, the proof is the second post in the thread there.
http://www.physicsforums.com/showthread.php?t=184502