# How to solve this differential equation (Logistic derivation)

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.

• It is a Separable Equation, so separate, integrate, simplify. – Moo Mar 26 '17 at 2:24

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?