Chapter 3.1, Problem 39E

(a)

To determine

To solve: The initial value problem $\frac{dy}{dt}=-r\left(1-\frac{y}{A}\right)y$, $y\left(0\right)=y_0$.

(b)

To determine

The value of ${\lim }_{t\to \infty }y\left(t\right)$ if $y_0<A$.

(c)

To determine

To show: That the value of ${\lim }_{t\to \infty }y\left(t\right)=\infty$, if $y_0>A$.

(d)

To determine

To show: That if $y_0>A$, then $y\left(t\right)$ has a vertical asymptote at $t=\frac{1}{r}\ln \left(\frac{y_0}{y_0-A}\right)$.