JEE Main · 2014 · 06 Apr 2014 · Offline · Differential Equations

Let the population of rabbits surviving at a time

t

be governed by the differential equation

\frac{d p ( t )}{d t} = \frac{1}{2} p (t) - 200

. If

p (0) = 100

, then

p (t)

equals

Answer: (A)

Solution

Write the equation as

\frac{d p}{d t} = \frac{1}{2} (p - 400)

and separate:

\frac{d p}{p - 400} = \frac{1}{2} d t

. Integrating gives

ln ∣ p - 400∣ = \frac{t}{2} + C

, and the start value

p (0) = 100

fixes

C = ln 300

. So

∣ p - 400∣ = 300 e^{t /2}

, and since the population begins below

400

400 - p = 300 e^{t /2}

. Hence

p (t) = 400 - 300 e^{t /2}

JEE Main · 2014 · 06 Apr 2014 · Offline · Single correct option

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