Given the following Cauchy problem
$$ \begin{cases} x'=x^m(x-1)^n(x-2)^{\frac{1}{3}}\\ x(t_0)=x_0 \end{cases} $$
Where $m,n \in \mathbb{N}$. For every $(t_0, x_0) \in \mathbb{R}^2$ Find the Maximal Interval of Existence.
I'm having a hard time finding a solution for $x(t)$, which makes me think I should use some other property of $x'$. I am aware that differentiating $x'$ will give me a way to find the interval where it's a locally lipschitz function, but I am not sure if this interval is the same as the Maximal Interval of Existence.