Chapter 9.3, Problem 26E

a.

To determine

To solve: The differential equation $\frac{dP}{dt}=cP\ln \left(\frac{L}{P}\right)$ with $P\left(0\right)=P_0$.

b.

To determine

The value of the limit $\underset{t\to \infty }{\lim }P\left(t\right)$.

c.

To determine

To show: The function $P\left(t\right)$ is increasing most rapidly when $P=\frac{L}{e}$.

d.

To determine

To show: The function $P\left(t\right)$ is increasing most rapidly when $t=\frac{\ln \left(\ln \left(\frac{L}{P_0}\right)\right)}{c}$.