Search any question & find its solution
Question:
Answered & Verified by Expert
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
Options:
Solution:
1008 Upvotes
Verified Answer
The correct answer is:
$\mathrm{y}=\mathrm{ce}^{\mathrm{kt}}$, for some constants $\mathrm{c} \geq 0$ and $\mathrm{k} \leq 0$
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 $\mathrm{k} \leq 0$
$\frac{\mathrm{dy}}{\mathrm{dt}} \propto \mathrm{y} \Rightarrow \frac{\mathrm{dy}}{\mathrm{dt}}=\mathrm{ky}$
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 $\mathrm{k} \leq 0$
Looking for more such questions to practice?
Download the MARKS App - The ultimate prep app for IIT JEE & NEET with chapter-wise PYQs, revision notes, formula sheets, custom tests & much more.