The population P of a town grows at a rate proportional to the population present, so that \(\displaystyle\frac{dP}{dt} = kP\), where k is a positive constant and t is the time. The general solution of this differential equation is (C is an arbitrary constant)
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → \(P = Ce^{kt}\)
The variables separate:
\( \displaystyle \frac{dP}{P} = k\,dt \)
Integrating both sides (P is positive):
\(\log P = kt + c\)
\(P = e^{kt + c} = e^{c}\,e^{kt}\)
Writing \(e^{c}\) as C:
\(P = Ce^{kt}\)
Why the other options are incorrect
Option (1) \(P = kt + C\) — Incorrect. Its rate is k, a constant, not kP.
Option (2) \(P = Ckt\) — Incorrect. Its rate is Ck, a constant, not kP.
Option (3) \(P = e^{kt} + C\) — Incorrect. Its rate is \(ke^{kt}\), which equals \(k(P - C)\), not kP.