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

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

The population of a rare species of bird on an isolated island is modelled by the differential equation

\frac{dP}{dt} = 0.08P\left(1 - \frac{P}{1200}\right)

where

P

is the population at time

t

years,

t \geq 0

(a)

State the conditions on

P

for the population to be increasing. [2 marks]

(b)

Given that the initial population is 300 birds, solve the differential equation to find an expression for

P

in terms of

t

. Give your answer in the form

P = \dfrac{A}{1 + Be^{-kt}}

, where

A

B

k

are constants. [3 marks]

(c)

Calculate the population after 10 years, correct to the nearest whole number. [1 mark]

Solutions

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