A-LevelMathematicsDifferential equations9709/31 · May/June 2022 · Q47 Marks

The variables x and y satisfy the differential equation `dy/dx = xy / (1 + x²)` and y = 2 when x = 0. Solve the differential equation, obtaining a simplified expression for y in terms of x.

The variables x and y satisfy the differential equation `dy/dx = xy / (1 + x²)` and y = 2 when x = 0. Solve the differential equation, obtaining a simplified expression for y in terms of x.[7]

Mark scheme

Marking points from the official mark scheme (May/June 2022). The mark or mark type is shown in brackets.

[B1] Separate variables correctly dy/y = x/(1+x²) dx Accept without integral signs.
[B1] Obtain term In y
[M1] State term of the form k ln(1+x²)
[A1] State correct term 1/2 ln(1+x²) OE
[M1] Evaluate a constant, or use limits x = 0, y = 2 in a solution containing terms a Iny and bln(1+x²) where ab≠0 If they remove logs first the constant must be of the correct form.
[A1] Obtain correct solution in any form e.g. lny + ln½= ln(1+x²)
[A1] Simplify and obtain y=2√(1+x²) OE The question asks for simplification, so need to deal with exp(ln(...)) .

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Topic: Differential equations, A-Level Mathematics (syllabus 9709). Structured question, 7 marks.
Source: A-Level Mathematics, May/June 2022 Paper 3 Variant 1, question 4.

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