All alternative solution for an equation I'm looking for all alternative solutions of this
$$x'=x(x-1)(x+1)$$
But I absolutely don't know what I have to do!
Thanks!

Let we start by noticing that
$$\frac{1}{(z-1)z(z+1)} = \frac{1/2}{z-1}-\frac{1}{z}+\frac{1/2}{z+1}\tag{1}$$
by partial fraction decomposition/the residue theorem. It follows that the separable DE
$$ \frac{x'}{(x-1)x(x+1)} = 1 \tag{2}$$
leads to
$$ \frac{1}{2}\log(x-1)-\log(x)+\frac{1}{2}\log(x+1) = t+C \tag{3} $$
or to:
$$ \log\left(1-\frac{1}{x^2}\right) = 2t+D\tag{4} $$
from which:
$$ x(t)^2 = \color{red}{\frac{1}{1-K e^{2t}}}\tag{5} $$
where $K>0$ depends on the initial conditions. In particular, $x(0)$ tells us exactly where the associated solution has a blowup.
