Q59 / 120 · VITEEE 2008 · Mathematics
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$\mathrm{y}=\mathrm{e}^{\mathrm{kt}}+\mathrm{c}$, for some constants $\mathrm{c} \leq 0$ and $\mathrm{k} \geq 0$
- Correct answerB$\mathrm{y}=\mathrm{ce}^{\mathrm{kt}}$, for some constants $\mathrm{c} \geq 0$ and $\mathrm{k} \leq 0$
- C$\mathrm{y}=\mathrm{e}^{\mathrm{ct}}+\mathrm{k}$, for some constants $\mathrm{c} \leq 0$ and $\mathrm{k} \geq 0$
- D$y=k e^{\mathrm{ct}}$, for some constants $\mathrm{c} \geq 0$ and $\mathrm{k} \leq 0$
Solution
According to the question,
$\frac{\mathrm{dy}}{\mathrm{dt}} \propto \mathrm{y} \Rightarrow \frac{\mathrm{dy}}{\mathrm{dt}}=\mathrm{ky}$
Separating the variables, we get $\frac{\mathrm{dy}}{\mathrm{dt}}=\mathrm{kdt}$
Integrating both sides, we get $\int \frac{\mathrm{dy}}{\mathrm{y}}=\int \mathrm{k} \mathrm{dt}$
$\log y=k t+M($ as $y$ cannot $b e-v e)$
$\Rightarrow \mathrm{y}=\mathrm{e}^{\mathrm{kt}+\mathrm{M}} \Rightarrow \mathrm{y}=\mathrm{e}^{\mathrm{M}} \cdot \mathrm{e}^{\mathrm{kt}}$
$\mathrm{y}=\mathrm{C} \mathrm{e}^{\mathrm{kt}}$, where $\mathrm{C}=\mathrm{e}^{\mathrm{M}}$
Constant $\mathrm{k}$ cannot be positive because the population never increases in time. And another constant $\mathrm{C}$ cannot be negative because of $\mathrm{e}^{\mathrm{M}}>0$ always.
Hence $y=\mathrm{Ce}^{\mathrm{kt}}$, for some constants $\mathrm{C} \geq 0$ and $\mathrm{k} \leq 0$
Hence, the correct answer is option: 'B'.