Understanding Uniqueness and Existence Theorems for ODE's

[manimaran1605]

How to understand Uniqueness and existence theorem for first order and second order ODE's intuitively?

 

[Strum]

I guess you should detail your question some more. What don't you understand and so forth. The wiki page for first order equations has a section for intuition
http://en.wikipedia.org/wiki/Picard–Lindelöf_theorem#Intuitive_understanding_of_the_theorem

 

[WWGD]

Following up on Strum's comment, it is a corollary of the contraction mapping theorem, a.k.a, Banach fixed point theorem.

 

[homeomorphic]

It's possible to prove existence and uniqueness using Euler's method, so if you understand Euler's method, that gives you some insight. But the actual proof that it works is kind of nasty--at least the one that I saw.

It's basically an iterated mapping from the set of smooth curves to itself that's a contraction mapping, so it has a fixed point that it goes towards, which is the solution curve. If you look carefully at the Picard iterations, it is possible to picture what they are doing. It's integrating all vectors that lie along the previous curve to get the next curve. So, for example, if you started with a stationary curve and there is a non-zero vector there, it will be corrected because it will move in the direction of that vector. The solution curve is the one that gives itself back when this procedure is applied.

Euler's method is a bit easier to understand intuitively.