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. \(y=ekt+c,\) for some constants \(c\le 0\) and \(k\ge 0\)
• B. \(y=cek^t,\) for some constants \(c\ge 0\) and \(k\le 0\)
• C. \(y=ect+k,\) for some constants \(c\le 0\) and \(k\ge 0\)
• D. \(y=ke^{ct},\) for some constants \(c\ge 0\) and \(k\le 0\)

Hint:

If \(\frac{dy}{dt}=ky\), solution is \(y=Ce^{kt}\). For decreasing population, \(k<0\).

Answer 1

The Correct Option is B

Step 1: Translate statement into differential equation.
Rate of change proportional to population: 
\[ \frac{dy}{dt}\propto y \Rightarrow \frac{dy}{dt}=ky \] 
Step 2: Solve the differential equation. 
\[ \frac{dy}{y}=k\,dt \] 
Integrate: 
\[ \ln|y|=kt+C \] 
\[ y=Ce^{kt} \] 
Step 3: Use the condition that population never increases. 
Population never increases \(\Rightarrow \frac{dy}{dt}\le 0\). 
Since \(y>0\), this implies: 
\[ k\le 0 \] 
Step 4: Identify correct form among options. 
The form \(y=Ce^{kt}\) with \(C\ge 0\) and \(k\le 0\) matches option (B). 
Final Answer: 
\[ \boxed{\text{(B) } y=Ce^{kt}\text{ where }C\ge 0,\ k\le 0} \]