A-LevelMathematicsDifferential equations9709/32 · Feb/Mar 2020 · Q68 Marks

The variables x and y satisfy the differential equation dy/dx = (1+4y²) / e^(x²). It is given that y = 0 when x = 1.

(a)Solve the differential equation, obtaining an expression for y in terms of x.[7]
(b)State what happens to the value of y as x tends to infinity.[1]

Mark scheme

Marking points from the official mark scheme (Feb/Mar 2020). The mark or mark type is shown in brackets.

[B1] Separate variables correctly and attempt integration of at least one side
[M1] Obtain term of the form atan⁻¹(2y)
[A1] Obtain term 1/2tan⁻¹(2y)
[B1] Obtain term -e⁻ˣ
[M1] Use x = 1, y = 0 to evaluate a constant or as limits in a solution containing terms of the form atan⁻¹(by) and ce⁻ˣ
[A1] Obtain correct answer in any form
[A1] Obtain final answer y = 1/2tan(2e⁻¹ - 2e⁻ˣ), or equivalent
[B1FT] State that y approaches 1/2tan(2e⁻¹), or equivalent (The FT is on correct work on a solution containing e⁻ˣ)

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Topic: Differential equations, A-Level Mathematics (syllabus 9709). Structured question, 8 marks.
Source: A-Level Mathematics, Feb/Mar 2020 Paper 3 Variant 2, question 6.

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