A-LevelMathematicsDifferential equations9709/33 · Oct/Nov 2025 · Q108 Marks
Express 2/(1-9y²) in partial fractions.
- (a)Express 2/(1-9y²) in partial fractions.[2]
- (b)The variables x and y satisfy the differential equation 2 cos²3x(dy/dx) = 1-9y², and y = 0 when x = (1/12)π. Solve the differential equation and obtain an expression for y in terms of x.[6]
Mark scheme
Marking points from the official mark scheme (Oct/Nov 2025). The mark or mark type is shown in brackets.
- Carry out a relevant method to find A and B such that 2/(1-9y²) = A/(1+3y) + B/(1-3y)
- Obtain A = 1 and B = 1
- Separate variables correctly and attempt integration of at least one side
- Integrate 2/(1-9y²) to obtain (1/3)ln(1+3y) - (1/3)ln(1-3y)
- Obtain rtan 3x
- Obtain term (1/3)tan 3x
- Use y = 0 when x = (1/12)π to evaluate a constant or as limits in a solution of the form p ln(1+3y)+q ln(1-3y)+r tan 3x where p,q,r ≠ 0
- Obtain answer y = (e^(tan3x-1)-1) / (3(1+e^(tan3x-1))) or y = (e^(tan3x-1)-1) / (3(e^(tan3x-1)+1))
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