The population p(t) at time t of a certain mouse species satisfies the differential equation \frac{dp \left(t\right)}{dt} = 0 . 5 p \left(t\right) - 450 . If p(0) = 850, then the time at which the population becomes zero is :

Engineering

Mathematics

Linear Differential Equation

Question

The population p(t) at time t of a certain mouse species satisfies the differential equation

$\frac{dp (t)}{dt} = 0.5 p (t) - 450$ . If p(0) = 850, then the time at which the population becomes zero is :

ln 9

ln 18

2 ln 18

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

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$\frac{dp (t)}{dt} = \frac{1}{2} p (t) - 450$

$∴ If e^{\int \frac{- 1}{2} dt} = e^{- t / 2}$

$∴ p (t) e^{- t / 2} = - 450 \int e^{- t / 2} dt + k$

$∴$ p(t) = 900 + ke^t/2 .......(A)

When t = 0 ; P(0) = 850

$∴$ k = – 50

$∴$ Equation (A) becomes

p(t) = 900 – 50et^/2

\ when p(t) = 0 then t = 2ln 18.