Interval of existence / uniqueness

[JJBladester]

Homework Statement

Problem 1 of 2:
Why is it that the continuity of a function in a region R and the continuity of the first partial derivative on R enables us to say that not only does a solution exist on some interval I₀, but it is the only solution satisfying y(x₀) = y₀?

Problem 2 of 2:
Explain why two different solution curves cannot intersect or be tangent to each other at a point (x₀,y₀) in R.

Homework Equations

Existence of a unique solution

The Attempt at a Solution

For Problem 1, I have no clue.

For Problem 2, I am assuming that the answer is simple: it is impossible for any single point in space to have more than one tangent line (slope), thus two different solution curves cannot intersect or be tangent at a specific point within region R.

 

[Dick]

The solution to what? What is the problem?

 

[JJBladester]

The solution to what? What is the problem?

I understand the vagueness of my post... The "problems" aren't really problems. I have read through the first chapter of my Diff Eq book and am stuck on some basics.

Primarily, I'm stuck on the proof and understanding behind the existence/uniqueness theorum and the reason why solution curves cannot intersect or be tangent at a single point.

I am taking this course completely on my own as I am unable to make it to class. I have been seeking out help online and through friends. I also have a kind-hearted friend in class who is taking notes for me and scanning/e-mailing them. Aside from that and MIT's Open CourseWare video lectures, would you have any other good pointers on getting BASIC information relating to Diff Eq? Any good sites or resources (especially for the two questions I initially posted)?