The variables x and y satisfy the differential equation dy/dx = 4 cos² y tan x, for 0 ≤ x < ½π, and x = 0 when y = ¼π. Solve this differential equation and find the value of x when y = ¾π.

Mark scheme

Marking points from the official mark scheme (Oct/Nov 2017). The mark or mark type is shown in brackets.

1. [B1] Separate variables correctly and attempt integration of one side
2. [B1] Obtain term tan y, or equivalent
3. [M1] Obtain term of the form k ln cos x, or equivalent
4. [A1] Obtain term -4 ln cos x, or equivalent
5. [M1] Use x = 0 and y = ¼π in solution containing a tan y and b ln cos x to evaluate a constant, or as limits
6. [A1] Obtain correct solution in any form, e.g. tan y = 4 ln sec x + 1
7. [M1] Substitute y = ¾π in solution containing terms a tan y and b ln cos x, and use correct method to find x
8. [A1] Obtain answer x = 0.587