Chapter 9.4, Problem 17E

To determine

(a)

To show:

The differentiation of $\frac{dy}{dt}=ky\left(1-\frac{y}{A}\right)$ gives $\frac{d^{2}y}{d{t}^2}=k^{2}y\left(1-\frac{y}{A}\right)\left(1-\frac{2y}{A}\right)$.

To determine

(b)

To show:

Graph of the function y is concave up if $0<y<\raisebox{1ex}{$A$}\!\left/ \!\raisebox{-1ex}{$2$}\right.$ and concave down if $\raisebox{1ex}{$A$}\!\left/ \!\raisebox{-1ex}{$2$}\right.<y<A$.

To determine

(c)

To show:

If $0<y(0)<\raisebox{1ex}{$A$}\!\left/ \!\raisebox{-1ex}{$2$}\right.$ then y has a point of inflection at $y=\frac{A}{2}$.

To determine

(d)

To find:

The time t where $y\left(t\right)$ reaches the inflection point if $0<y(0)<\raisebox{1ex}{$A$}\!\left/ \!\raisebox{-1ex}{$2$}\right.$.