Uniqueness of solutions – Picard Theorem.

Question

We have the IVP:
\begin{equation*}
\left\{
\begin{array}{ll}
y'(x)=\sqrt[3]y & \quad \\
y(0)=0 & \quad
\end{array}
\right.
\end{equation*}
By Peano's Theorem, we know a solution exists in a rectangle. You can also solve this EDO getting $y(x)=\dfrac{2x}{3}^{\frac{3}{2}}$.
Since it isn't Lipschitz-Continuous, we can't use Picard's Theorem to ensure uniqueness.
I can't find find two solutions to show it isn't unique, any hint?

Answer 1

The function $$y(x)=0$$ also satisfies your initial value problem.

$$\begin{equation*} \left\{ \begin{array}{ll} y'(x)=\sqrt[3]y & \quad \\ y(0)=0 & \quad \end{array} \right. \end{equation*}$$