JEE Main · 2012 · 29 Apr 2012 · Offline · Differential Equations

The population

p (t)

at time

t

of a certain mouse species satisfies the differential equation

\frac{d p ( t )}{d t} = 0.5 p (t)

-450 . If

p (0) = 850

, then the time at which the population becomes zero is

Answer: (A)

Solution

Rewrite

\frac{d p}{d t} = 0.5 p - 450

\frac{d p}{d t} = \frac{p - 900}{2}

. Separating variables,

2 \int \frac{d p}{p - 900} = \int d t

, which integrates to

2 ln ∣ p - 900∣ = t + C

. The start

p (0) = 850

fixes

C = 2 ln 50

, since

∣850 - 900∣ = 50

. Setting

p = 0

gives

2 ln 900 = t + 2 ln 50

, so

t = 2 ln \frac{900}{50} = 2 ln 18

JEE Main · 2012 · 29 Apr 2012 · Offline · Single correct option

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