6.14 Solution of the logistic equation

The logistic equation is separable: we can solve it by dividing both sides through by $N(1-N/K)$. Then we obtain:

Using partial fractions, the left-hand side can be expressed in the form $\frac{A}{N}+\frac{B}{K-N}$. Then we obtain $A=B={1.}$ Thus the left-hand side is

which equals $rt+c$ for some constant $c$. Taking the exponential of both sides, we obtain $\frac{N}{K-N}=\pm Ce^{rt}$ for a positive constant $C$. Rearranging the equation, we obtain $N(t)={\frac{K}{1+C^{\prime}e^{-rt}}}$ for some non-zero constant $C^{\prime}$.