I'm currently trying to solve the differential equation $$\frac{dN}{dt} = rN\bigg(1-\big(\frac{N}{k}\big)^2\bigg)$$ Where N is the population of a fish and r, k are positive constants.
I've tried rearranging and using the substitution $u = N^{-2}$ but i'm not having much luck due to the two separate terms on the RHS. ie.
Using the substitution, $du = -\frac{dN}{N}$ and $u = N^{-2}$.
$$\int -u \cdot du = \int (ru - \frac{r}{k^2})\cdot dt$$
Is the furthest I can make it!