Source: https://tutorial.math.lamar.edu/Classes/DE/IoV.aspx, Example 2.
Determine all possible solutions to the following IVP. $$y'=y^{\frac{1}{3}}, \space\space\space\space\space\space\space\ y(0)=0$$
To solve this ODE the author divides both sides by $y^{\frac{1}{3}}$. This seems wrong to me. We want a solution in a neighbourhood of $x=0$, and $y^{\frac{1}{3}}(0)=0$ therefore we would be dividing by $0$ at that point.
Your observation is correct, that's why you at first examine the zero function and find that it is a solution.
Now the right side is not Lipschitz at $y=0$, so you have to consider the possibility of other solutions. Thus assume that $y(t)\ne 0$ for $t>0$. Then the division is not by zero at points $t>0$.
A similar treatment can be done for $t<0$, it turns out that there are no solutions but the zero solution that end up in $y(0)=0$ on the negative half-axis.