Question

Determine whether Theorem 1 does or does not guarantee existence of a solution of the given initial value problem. If existence is guaranteed, determine whether Theorem 1 does or does not guarantee uniqueness of that solution.

dy/dx = 2x²y²       y(1) = −1.

Answer 1

The function f(x, y) is:

f(x, y) = 2x²y²

Its partial derivative with respect to y is:

∂f/∂y = 4x²y.

 Both f(x, y) and ∂f/∂y are continuous everywhere. So, there exists a unique solution around any initial value (a, b), including (1, −1).