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Unit 13 Differential Equations
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Unit 13 Differential Equations
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Maths Department A Level MathsUnit 13 Book 2
Differential Equations Assessed Homework
1. A car is bought for £25 000 on the 1st June 2017
The value, V thousand pounds, of the car decreases at a rate proportional to V.
(a) Construct a differential equation involving V, t and a positive constant k, to model
this situation, where t is the number of years after 1st June 2017.
[3 marks]
On the 1st June 2019, the car is worth £15 000
(b) Find the value of the car on the 1st June 2022
[7 marks]
2. Solve the differential equation
𝑑𝑦 6𝑥𝑒 3𝑥
=
𝑑𝑥 𝑦2
given that 𝑦 = 2 when 𝑥 = 0.
Give your answer in the form 𝑦 3 = 𝑓(𝑥).
[8 marks]
3. (a) Prove that
𝑑 4 𝜋
(ln tan 2𝑦) = , 0<𝑦<
𝑑𝑦 sin 4𝑦 4
[4 marks]
(b) Solve the differential equation
𝑑𝑦
= 2 cos 𝑥 sin 4𝑦
𝑑𝑥
𝜋
given that 𝑦 = when 𝑥 = 0.
6
Give your answer in the form tan 2𝑦 = 𝐴𝑒 𝐵 sin 𝑥 , where 𝐴 and 𝐵 are constants to be
determined.
[6 marks]
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