Chapter 8.5, Problem 46E

(a)

To determine

To graph: The possible solution of the differential equation $\frac{dy}{dt}=3\left(3-y\right)$ on the provided slope field (Refer to the figure provided in the question) when $y\left(0\right)=5$ and another when $y\left(0\right)=\frac{1}{2}$.

(b)

To determine

The sign of the slope of the solution of the differential equation $\frac{dy}{dt}=3\left(3-y\right)$, where $y\left(0\right)$ is greater than $3$.

(c)

To determine

The value of the limit $\underset{t\to \infty }{\lim }y\left(t\right)$, for $y>0$ for the solution of the differential equation $\frac{dy}{dt}=3\left(3-y\right)$ using the provided slope filed (Refer to the figure provided in the question).

(d)

To determine

To calculate: The integral ${\displaystyle \int \frac{dy}{y\left(L-y\right)}}={\displaystyle \int kdt}$ and solve for $y$ as a function of $t$.

(e)

To determine

To calculate: The solution of the differential equation $\frac{dy}{dt}=3\left(3-y\right)$ when $y\left(0\right)=5$ and another when $y\left(0\right)=\frac{1}{2}$ using the result obtained in part (d) and to graph them.

(f)

To determine

To prove: The rate of growth is maximum at the point of inflection and that occurs at $y=\frac{L}{2}$ for the differential equation $\frac{dy}{dt}=ky\left(L-y\right)$.