Infinite many solutions of the differential equation. Consider the differential equation $\frac{dy}{dx} =y^{1/3} $ with $y(0)=0$. Then show that the given differential equation has infinite many solutions.
I found two solutions of the given equation $y_{1}$ and $y_{2}$ where $y_{1}=0$ that is zero and
$$y_{2}=
\left\{
 \begin{array}{ll}
  (\frac{2}{3}x)^{3/2}  & \mbox{if } x \geq 0 \\
  0 & \mbox{if } x \leq 0
 \end{array}
\right. $$
Please help me to find other solution's of the  given differential equation.

The solution you have will give more solutions when shifted:
$$y=\cases{(\frac23(x-C))^{3/2}&if $x\ge C$\cr 0&if $x<C$\cr}$$
satisfies the differential equation.  Provided $C\ge0$, the initial condition is also satisfied.
