This is a standard example in connection with uniqueness of solutions to initial value problems. The initial value problem $y'=3y^{2/3}\ ,\quad y(0)=0$ does not satisfy the essential technical assumption of the existence and uniqueness theorem, because $\lim_{y\to0}{|y|^{2/3}\over |y|}=\infty\ .$ Therefore we cannot expect a unique solution. As other contributors have noted the functions $x\to0 \ (x\in{\mathbb R})$ and $x\to x^3 \ (x\in{\mathbb R})$ are solutions; and as the differential equation is "$x$-free" an infinity of further solutions can be "spliced" using these and their translates.
In connection with initial problems it is reasonable to consider two solutions as one and the same if they coincide in a neighborhood of the initial point (proceeding to the respective equivalence classes one then talks about germs of solutions). In our example there are exactly four different germs.
Note that the phenomenon observed here is not pathological. It turns up whenever we have an envelope to a given family of curves.