The variables x and y satisfy the differential equation cos 2x dy/dx = 4 tan 2x / sin² 3y, where 0 ≤ x < π/4. It is given that y = 0 when x = π/6. Solve the differential equation to obtain the value of x when y = π/6. Give your answer correct to 3 decimal places.

Mark scheme

Marking points from the official mark scheme (May/June 2023). The mark or mark type is shown in brackets.

1. [B1] Correct separation of variables ∫sin² 3y dy = ∫4sec2x tan2xdx or equivalent. Condone missing integral signs or dx and dy.
2. [M1] Integrate to obtain k sec 2x
3. [A1] Obtain 2 sec 2x
4. [M1] Use double angle formula and integrate to obtain py + q sin6y Or two cycles of integration by parts.
5. [A1] Obtain ½ y - 1/12 sin 6y
6. [M1] Use y=0, x=π/6 in a solution containing terms λsec2x and μsin6y to find the constant of integration
7. [A1] Obtain ½ y - 1/12 sin 6y = 2 sec 2x - 4 Or equivalent seen or implied by ½(π/6) - 1/12 sin(π) = 2sec(π/3) - 4.
8. [A1] Obtain x = 0.541 From correct working (not by using the calculator to integrate).