QJEEM12Ma9520: JEE Main 2012 Mathematics - Class 12 JEEM12Ma9520 Solution: The population at time of a certain mouse species satisfies the …

Question

Verified Solution • 2025–26 Session moderate

The population

p

(t)

at time

t

of a certain mouse species satisfies the differential equation

{{dp\left( t \right)} \over {dt}} = 0.5\,p\left( t \right) - 450.\,\,

If

p(0)=850,

then the time at which the population becomes zero is :

Accepted Answer

Given differential equation is

{{dp\left( t \right)} \over {dt}} = 0.5p\left( t \right) - 450

\Rightarrow {{dp\left( t \right)} \over {dt}} = {1 \over 2}p\left( t \right) - 450

\Rightarrow {{dp\left( t \right)} \over {dt}} = {{p\left( t \right) - 900} \over 2}

\Rightarrow 2{{dp\left( t \right)} \over {dt}} = \left[ {900 - p\left( t \right)} \right]

\Rightarrow 2{{dp\left( t \right)} \over {900 - p\left( t \right)}} = - dt

Integrate both sides, we get

\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, - 2\int {{{dp\left( t \right)} \over {900 - p\left( t \right)}}} = \int {dt}

\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,Let\,\,900 - p\left( t \right) = u

\Rightarrow - dp\left( t \right) = du

\therefore

\,\,\,\,\,\,

We have,

2\int {{{du} \over u}} = \int {dt \Rightarrow 2\,\ln \,u = t + c}

\Rightarrow 2\ln \left[ {900 - p\left( t \right)} \right] = t + c

\,\,\,\,\,\,\,\,\,\,\,\,\,\,

when

t = 0,p\left( 0 \right) = 850

\,\,\,\,\,\,\,\,\,\,\,\,\,\,

2\ln \left( {50} \right) = c

\Rightarrow 2\left[ {\ln \left( {{{900 - p\left( t \right)} \over {50}}} \right)} \right] = t

\Rightarrow 900 - p\left( t \right) = 50{e^{{t \over 2}}}

\Rightarrow p\left( t \right) = 900 - 50{e^{{t \over 2}}}

Let

p\left( {{t_1}} \right) = 0

0 = 900 - 50{e^{{{{t_1}} \over 2}}}

\therefore

\,\,\,\,\,\,{t_1} = 2\ln 18

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