Initial value problem $dy/dx=y^{1/3}$, $y(0)=0$ has one of the following solution

I came across the following problem which says:

The initial value problem $y'=y^{1/3}$, $y(0)=0$ has:

(a) a unique solution,

(b) exactly two solutions,

(c)exactly three solutions,

(d)no solution.

The solution of the problem is given by: $2x=3y^{2/3}$. But I could not come to a conclusion. Clearly, (d) can not be true. But i am not sure about the other options. It will be helpful if someone throws light on it. Thanks in advance.

Solution 1:

The answer is "none of the above". There are infinitely many solutions.

Pick any $\alpha > 0$ and define $f_\alpha (x) = 0$ for $x \le \alpha$ and $f(x) = (2/3)^{3/2} (x-\alpha)^{3/2}$ for $x > \alpha$. All these functions are solutions.