Chapter 8.2, Problem 49ES

Assume that $y=y\left(t\right)$ satisfies the logistic equation with $y_0=y\left(0\right)$ the initial value of y.

(a) Use separation of variables to derive the solution

$y=\frac{y_{0}L}{y_0+\left(L-y_0\right)e^{-kt}}$

(b) Use part (a) to show that $\underset{t\to +\infty }{\lim }y\left(t\right)=L.$