Let $k \in R$, consider the following Cauchy problem
$$y'(x) = y(x)(y(x)-1)^{1/3} $$ $$y(0) = k $$
To draw the graph of solutions, defining the domain, studying the monotonicity, the convexity, and limits at the extrema of the domain for
$1)k<0$
$2)k\in(0,1)$
$3)k>1$
$4)k=1$
I don't understand how to proceed after knowing the fact that $y =0 $ and $y = 1$ are the only constant solutions and these cannot be crossed given the uniqueness. However, I have the solution yet I am facing the problem of understanding key concepts. I have attached the link below to get a look. I have tried solving this using the slope-field method but it came out very wrong. I request anyone who can discuss or provide a slightest hint to solve this.
https://mate.unipv.it/rocca/Solutions-Adv.Math.Meth.January25-2022.pdf