A-LevelMathematicsDifferential equations9709/33 · May/June 2018 · Q68 Marks
6
- (i)Express 1/(4-y²) in partial fractions.[2]
- (ii)The variables x and y satisfy the differential equation x (dy/dx) = 4-y², and y = 1 when x = 1. Solve the differential equation, obtaining an expression for y in terms of x.[6]
Mark scheme
Marking points from the official mark scheme (May/June 2018). The mark or mark type is shown in brackets.
- M1 Carry out relevant method to find A and B such that 1/(4-y²) = A/(2+y) + B/(2-y)
- A1 Obtain A = B = 1/4
- M1 Separate variables correctly and integrate at least one side to obtain one of the terms a ln x, b ln (2 + y) or c ln (2 −y)
- B1 Obtain term ln x
- A1FT Integrate and obtain terms (1/4)ln(2+y) - (1/4)ln(2-y)
- M1 Use x = 1 and y = 1 to evaluate a constant, or as limits, in a solution containing at least two terms of the form a ln x, b ln (2 + y) and c ln (2 – y)
- A1 Obtain a correct solution in any form, e.g. ln x = (1/4)ln(2+y) - (1/4)ln(2-y) - (1/4)ln3
- A1 Rearrange as y = (2(3x^4-1))/(3x^4+1), or equivalent
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