(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

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 I_{0}, but it is the only solution satisfying y(x_{0}) = y_{0}?

Problem 2 of 2:

Explain why two different solution curves cannot intersect or be tangent to each other at a point (x_{0},y_{0}) in R.

2. Relevant equations

Existence of a unique solution

3. 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.

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# Interval of existence / uniqueness

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