Chapter 9.1, Problem 67E

Rumor spread–Gompertz growth model. The rate of propagation of a rumor is assumed to satisfy the Gompertz growth model

$\frac{dN}{dt}=N{e}^{-0.5t}$

where N(t) is the number of individuals who have heard the rumor at time t.

(A)    Show that $N\left(t\right)=C{e}^{-2e^{-0.5t}}$ is the general solution of this differential equation, and evaluate $\bar{N}=\underset{t\to \infty }{\lim }N\left(t\right)$.

(B)    Graph the particular solutions that satisfy N(0) = 100 and N(0) = 200.

(C)    Discuss the effect of the value of N(0) on the long-term propagation of this rumor.