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 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. 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.