# Existence And Uniqueness Theorem Question

(a) Does the existence and uniqueness theorem guarantee the uniqueness of the solution of the initial value problem

$dy/dx = 2x(y-2)^\frac{2}{3}, y(1) = 2$

Attempt:

NO because $∂/∂y = \frac{4x}{3 \sqrt[3]{y-2}}$

At (2,1), it is not continuous.

(b) Find the general solution of the equation. Use it to determine if the solution of the initial value problem is unique.

Attempt:

General Solution: $y = (\frac {x^2 + C}{3})^3 + 2$ or y = 2

If y(1) = 2, we get the solution $y = (\frac{x^2}{3})^3 +2$ or y = 2. Therefore not uniwue