A-LevelMathematicsDifferential equations9709/33 · Oct/Nov 2019 · Q910 Marks

The variables x and t satisfy the differential equation 5 dx/dt = (20−x)(40 – x). It is given that x = 10 when t = 0.

(i)Using partial fractions, solve the differential equation, obtaining an expression for x in terms of t.[9]
(ii)State what happens to the value of x when t becomes large.[1]

Mark scheme

Marking points from the official mark scheme (Oct/Nov 2019). The mark or mark type is shown in brackets.

[B1] Separate variables correctly and integrate one side
[B1] Obtain term 0.2t, or equivalent
[*M1] Carry out a relevant method to obtain A and B such that 1/((20-x)(40-x)) = A/(20-x) + B/(40-x)
[A1] Obtain A = 1/20 and B = -1/20
[A1FT] Integrate and obtain terms -1/20 ln(20-x) + 1/20 ln(40-x) OE
[DM1] Use x = 10, t = 0 to evaluate a constant, or as limits
[A1] Obtain correct answer in any form
[A1] Obtain final answer x = (60e^(4t) - 40) / (3e^(4t) - 1)
[B1] State that x approaches 20

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Topic: Differential equations, A-Level Mathematics (syllabus 9709). Structured question, 10 marks.
Source: A-Level Mathematics, Oct/Nov 2019 Paper 3 Variant 3, question 9.

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