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Differential Equations · Paper 2 — Free Maths AA SL Practice Questions

1MasterySAQ-SSolving first-order differential equationss6 marksPaper 2~9 min

A population of rabbits on an island is modelled by the differential equation

\frac{dP}{dt} = \frac{P(500 - P)}{1000}

where

P

is the number of rabbits at time

t

years,

t \geq 0

(a)

Show that the population is increasing when $0 < P < 500$. [2 marks]

(b)

Given that

P = 100

when

t = 0

, use the result

\int \frac{1}{P(500-P)}\,dP = \frac{1}{500}\ln\left|\frac{P}{500-P}\right| + C

to show that

P = \dfrac{500}{1 + 4e^{-t/2}}

. [4 marks]

Solutions

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