Question

Let $y$ be the number of people in a village at time $t$. Assume that the rate of change of the population is proportional to the number of people in the village at any time and further assume that the population never increases in time. Then the population of the village at any fixed time $t$ is given by

• A. $y=e^{kt}+c$, for some constants $c\le 0$ and $k\ge 0$
• B. $y=c{e}^{kt}$, for some constants $c$ $\ge$ 0 and k $\le$ 0
• C. $y=c{e}^{kt}$, for some constants $c\le 0$ and $k\ge 0$
• D. $y=k{e}^{ct}$, for some constants $c\ge 0$ and $k\le 0$

Answer 1

Answer: (B)

According to the question,
$\frac{dy}{dt}\propto y\Rightarrow \frac{dy}{dt}=ky$
Separating the variables, we get $\frac{dy}{dt}$ = kdt
Integrating both sides, we get $\int \frac{dy}{y}=\int kdt$
log y = k t + M (as y cannot be -ve)
$\Rightarrow y=e^{kt+M}\Rightarrow y=e^M.e^{kt}$
y$=C{e}^{kt},$ where C = e${}^M$
Constant k cannot be positive because the population never increases in time. And another constant C cannot be negative because of e${}^M$ > 0 always.
Hence y = Ce${}^{kt}$, for some constants C $\ge$ 0 and k $\le$ 0.