First slide

Methods of solving first order first degree differential equations

Question

$The population p(t) at time ‘t’ of a certain mouse species satisfies$
$differential equation \frac{d}{dt} p (t) = 0.5 p (t) - 450 . If P (0) = 850, then the time at which the population becomes zero is$

Moderate

A

$2 \ln 18$
B

$\ln 9$
C

$\frac{1}{2} \ln 18$
D

$\ln 18$

Solution

$\frac{d}{dt} p (t) = 0.5 p (t) - 450 \frac{d p (t)}{d t} = \frac{p (t)}{2} - 450$
$\begin{array}{l} 2 \int \frac{dp (t)}{900 - p (t)} = - \int dt \\ \Rightarrow - 2 \ln (900 - p (t)) = - t + c \\ W h e n t = 0, p (0) = 850 \Rightarrow c = - \ln (50) \\ - 2 \ln (900 - p (t)) = - t + [- \ln (50)] \\ ∴ 2 \ln (\frac{900 - p (t)}{50}) = t \\ When population becomes zero p (t) = 0 \\ 2 \ln \frac{900}{50} = t \\ 2 \ln 18 = t \end{array}$

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