Chapter 9.5, Problem 42E

(a)

To determine

To show: The differential equation $\frac{dP}{dt}=k\left(t\right)P\left(1-\frac{P}{M\left(t\right)}\right)$ from logistic differential equation $\frac{dz}{dt}+k\left(t\right)z=\frac{k\left(t\right)}{M\left(t\right)}$ by using $z=\frac{1}{P}$

(b)

To determine

To show: The differential equation $P\left(t\right)=\frac{M}{1+M{e}^{-{\displaystyle \int k\left(t\right)dt}}}$ for the expression $\frac{dz}{dt}+k\left(t\right)z=\frac{k\left(t\right)}{M\left(t\right)}$

(c)

To determine

To show: The function $z\left(t\right)=e^{-kt}{\displaystyle {\int }_0^t\frac{k{e}^{ks}}{M\left(s\right)}}ds+C{e}^{-kt}$ and deduce it using L’Hospital’s rule.