I’m trying to integrate the logistic equation, but I’m stuck with the partial fractions bit. None of the solutions I’ve found online (including here) clarify how it should go, and I keep getting something strange.
The form I‘ve arrived at after separating variables is:
$$\frac{dP}{P\left(1-\frac{P}{K}\right)}=r\,dt$$
And so I am trying to deal with the LHS of that:
$$\frac{dP}{P\left(1-\frac{P}{K}\right)} = \frac{A}{P} + \frac{B}{1-\frac{P}{K}}$$
The roots of $$-\frac{P^2}{K} + P = 0$$ are 0 and K. Thus,
$$dP = A(1-\frac{P}{K}) + BP$$
When I plug in the roots, I get A = 0 and B = d (?!)
So I must have a mistake somewhere, but I can’t find it. I’ll be grateful for any help!
UPD: Thank you for pointing out Mathjax to me. I’m sorry I didn’t format my solution properly before, I’m new here :)