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

1MasterySAQ-SSolving first-order differential equations6 marksPaper 2~9 min

The population of a colony of penguins on a remote island is modelled by the differential equation

\frac{dP}{dt} = 0.02P(1000 - P)

where

P

is the number of penguins at time

t

years after 1 January 2020. Initially, there are

200

penguins.

(a)

Show that

\frac{1}{P(1000-P)} = \frac{1}{1000}\left(\frac{1}{P} + \frac{1}{1000-P}\right).

[2 marks]

(b)

Hence, solve the differential equation to find an expression for

P

in terms of

t

. Give your answer in the form

P = \frac{a}{b + c\,e^{-kt}}

where

a,\, b,\, c,\, k \in \mathbb{N}

. [3 marks]

(c)

Calculate the number of penguins on 1 July 2020 (i.e. at

t = 0.5

), correct to the nearest whole number. [1 mark]

Solutions

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