A-LevelMathematicsDifferential equations9709/31 · Oct/Nov 2018 · Q57 Marks
The coordinates (x, y) of a general point on a curve satisfy the differential equation x dy/dx = (2 − x²)y. The curve passes through the point (1, 1). Find the equation of the curve, obtaining an expression for y in terms of x.
- (a)The coordinates (x, y) of a general point on a curve satisfy the differential equation x dy/dx = (2 − x²)y. The curve passes through the point (1, 1). Find the equation of the curve, obtaining an an expression for y in terms of x.[7]
Mark scheme
Marking points from the official mark scheme (Oct/Nov 2018). The mark or mark type is shown in brackets.
- [B1] Separate variables correctly and integrate at least one side
- [B1] Obtain term ln y
- [B1+B1] Obtain terms 2 ln x – 1/2 x²
- [M1] Use x = 1, y = 1 to evaluate a constant, or as limits
- [A1] Obtain correct solution in any form, e.g. ln y = 2 ln x – 1/2 x² + 1/2
- [A1] Rearrange as y = x² exp(1/2 – 1/2 x²) or equivalent
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