Chapter 9.5, Problem 33E

Solution of the logistic equation Use separation of variables to show that the solution of the initial value problem

     $\begin{array}{cc}{P}^{\prime }(t)=rP\left(1-\frac{P}{K}\right),& P(0)=P_0\end{array}$

 is $P(t)=\frac{K}{\left(\frac{K}{P_0}-1\right)e^{-rt}+1}$.