JEE PYQ: Differential Equation Question 8
Question 8 - 2021 (24 Feb Shift 1)
The population $P = P(t)$ at time ‘$t$’ of a certain species follows the differential equation $\frac{dP}{dt} = 0.5P - 450$. If $P(0) = 850$, then the time at which population becomes zero is:
(1) $\frac{1}{2}\log_e 18$
(2) $2\log_e 18$
(3) $\log_e 9$
(4) $\log_e 18$
Show Answer
Answer: (2)
Solution
$\frac{dP}{P-900} = \frac{1}{2}dt$. Integrating: $\ln|P-900| = \frac{t}{2} + C$. At $t = 0$: $\ln 50 = C$. For $P = 0$: $\ln 900 = \frac{t}{2} + \ln 50$, so $\frac{t}{2} = \ln 18$, $t = 2\ln 18$.
BETA
