The variables x and y satisfy the differential equation sin 4y dy/dx = x sin 2y sin 3x. It is given that y = π/6 when x = π/2.

Mark scheme

Marking points from the official mark scheme (May/June 2025). The mark or mark type is shown in brackets.

1. [B1] Separate variables correctly.
2. [M1*] Use correct double angle formula to simplify integral in y. sin 4y / sin 2y dy = integral 2 cos 2y dy.
3. [A1] Obtain sin 2y.
4. [M1*] Commence integration by parts and obtain pxcos3x+q integral cos3x dx.
5. [A1] Obtain -1/3 xcos3x + 1/3 integral cos3x dx.
6. [A1] Complete integration and obtain -1/3 xcos3x + 1/9 sin 3x.
7. [DM1] Use y = π/6 when x = π/2 in an expression with sin 2y, xcos3x and sin 3x to obtain the constant of integration. 1/2 = 0 - 0 + c.
8. [A1] Obtain sin 2y = -1/3 xcos3x + 1/9 sin 3x + 1/2. OE ISW.
9. [M1] Solve sin 2y = 1/2 to obtain one solution, e.g. 0.329. Allow for their constant from a solution involving sin 2y, x cos 3x and sin 3x. M0 for answers in degrees.
10. [A1] Obtain a second solution, e.g. 1.24, and no others in range. AWRT