Why Does ∂f/∂y Determine the Uniqueness of y in Differential Equations?

[MathewsMD]

Hi,

I was just wondering why taking ∂f/∂y provides the interval on which y is unique (or not necessarily). Could someone possibly provide some mathematical intuition behind this and possibly a proof of some sort detailing why y is unique if ∂f/dy is continuous? Also, how exactly (if it can) is uniqueness determined if ∂f/dy is discontinuous at a certain point?

 

[johnqwertyful]

I'm confused, what do you mean that "y is unique".

 

[lurflurf]

^Maybe this question is about The[/PLAIN] Picard–Lindelöf theorem?
link
http://en.wikipedia.org/wiki/Picard–Lindelöf_theorem

The theorem gives sufficient conditions so a solution that fails to satisfy the hypothesis might still be unique, but it would need to be shown by a different method. Of course there also might be multiple solutions.

There should be some discussion of this in most differential equations books.