A-LevelMathematicsDifferential equations9709/32 · Feb/Mar 2021 · Q47 Marks
The variables x and y satisfy the differential equation (1-cosx) dy/dx = y sinx. It is given that y = 4 when x = π.
- (a)Solve the differential equation, obtaining an expression for y in terms of x.[6]
- (b)Sketch the graph of y against x for 0 < x < 2π.[1]
Mark scheme
Marking points from the official mark scheme (Feb/Mar 2021). The mark or mark type is shown in brackets.
- [M1] Separate variables correctly and attempt integration of at least one side
- [A1] Obtain term ln y
- [M1] Obtain term of the form ±ln(1-cos x)
- [A1] Obtain term ln (1-cos x)
- [M1] Use x=π, y = 4 to evaluate a constant, or as limits, in a solution containing terms of the form a ln y and b ln(1-cos x)
- [A1 OE] Obtain final answer y = 2(1-cos x)
- [B1 FT] Show a correct graph for 0 < x < 2π with the maximum at x = π
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