https://www.ma.utexas.edu/users/davis/375/popecol/lec5/logist.html
How do I get from here:
$$\frac{dN}{dt}=rN=r_0 N\left(1-\frac{N}{K}\right)$$
To here:
$$N_t=\frac{N_0\cdot K}{N_0+(K-N_0)\cdot \exp(-r_0\cdot t)}$$
I know basic differential equations where you separate variables & integrate both sides, and add the $+C$, etc.
Well, this is a separable ODE, which you mentioned you know: $$\frac{dN}{dt}=r_0 N\left(1-\frac{N}{K}\right)$$ $$\int \frac{1}{N\left(1-\frac{N}{K}\right)}~dN=\int r_0~dt$$ $$\int \frac{K}{N(K-N)}~dN=\int r_0~dt$$ Can you integrate both sides (You can use partial fractions for the LHS), then substitute the initial condition (i.e. $N(0)=N_0$) to solve for the arbitrary constant?