Does there exist a unique solution for $y' = -y^{1/3}, y(0) = 0$?

Question

The given problem is to determine if the following initial value problems have a unique solution :

1. $y' = y^{\frac13}, y(0) = 0,$
2. $y' = -y^{\frac13}, y(0) = 0,$

We can not apply Picard's theorem here. For the first one, the answer in this post tells that solutions are infinite.

For the second one, I tried to construct a similar function given in the answer but was not successful. Since only the sign is different in the second one than the first one, I feel that the solutions are infinite but unable to prove or disprove them. Any help is higly appreciated.

Answer 1

The second equation also has infinitely many solutions, in a very similar form as the infinite family for the first. Let $a<0$, then $$y(x)=\begin{cases}(2/3)^{3/2}(\color{blue}{a-x})^{3/2}&x\le a\\0&x>a\end{cases}$$ is the desired infinite family.