How to solve the logistic equation for population, $y'=y(\frac{1}{K}-y)(\frac{1}{L}-y),y(0)=y_0$, where $K<L,K,L\in\mathbb{R}$?
I tried to expand the equation and move all the terms with $y$ to the LHS then use separable equations, but it didn't work. Can anyone give me an idea to start?
Write it as
$$\frac{dy}{dt}=y\left(\frac{1}{K}-y\right)\left(\frac{1}{L}-y\right)$$
So
$$\frac{1}{y\left(\frac{1}{K}-y\right)\left(\frac{1}{L}-y\right)}dy=dt$$
Where given $K,L\neq0\text{, }\frac{1}{K}\in\mathbb{R}$ and $\frac{1}{L}\in\mathbb{R}$ so you can call each of them a different name for simplicity like $a$ and $b$
$\left(\frac{1}{K}-y\right)\left(\frac{1}{L}-y\right)=\left(y-\frac{1}{K}\right)\left(y-\frac{1}{L}\right)$
and use partial fractions to expand