Step 1: The equation $\frac{dy}{dx}=2y^{1/3}$ has no explicit $x$ on the right side, so if $y(x)$ solves it, the shifted function $y(x-a)$ also solves it for any constant $a$, wherever it is defined.
Step 2: One explicit nontrivial solution is $y(x)=\left(\frac{4x}{3}\right)^{3/2}$ for $x\ge 0$, obtained from $\frac{3}{2}y^{2/3}=2x$.
Step 3: The trivial solution $y(x)=0$ is also a solution, and it matches the nontrivial branch in both value and slope (both zero) at $x=0$.
Step 4: Using translation invariance, glue the zero solution on $(-\infty,a]$ to the shifted nontrivial branch $\left(\frac{4(x-a)}{3}\right)^{3/2}$ on $(a,\infty)$, for any $a\ge 0$. The match in value and slope at $x=a$ keeps the combined curve differentiable everywhere.
Step 5: As $a$ ranges over $[0,\infty)$, this symmetry argument alone generates infinitely many distinct solutions of the same IVP.
\[\boxed{\text{Infinitely many solutions}}\]