Exercise 3

Consider the initial value problem $$\begin{aligned} y' & = 3y^{2/3} \\ y(0) & = 0. \end{aligned}$$

1. Show that the constant function, $y(t) \equiv 0$, is a solution to the initial value problem.

2. Show that $$y(t) = \begin{cases} 0, & t \leq t_0 \\ (t - t_0)^3, & t \gt t_0 \end{cases}$$ is a solution for the initial value problem, where $t_0$ is any real number. Hence, there exists an infinite number of solutions to the initial value problem.

3. Explain why this example does not contradict the Existence and Uniqueness Theorem.