The variables x and y satisfy the differential equation dy/dx = e³ʸ sin² 2x. It is given that y = 0 when x = 0. Solve the differential equation and find the value of y when x = ½.

Mark scheme

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

1. [B1] Separate variables correctly and obtain e⁻³ʸ and sin² 2x on the opposite sides
2. [B1] Obtain term -⅓e⁻³ʸ
3. [M1] Use correct double angle formula for sin²2x = (1/2)[1 – cos 4x]
4. [A1] Obtain terms ½x - ⅛ sin 4x oe
5. [M1] Use x = 0, y = 0 to evaluate a constant or as limits in a solution containing terms of the form ax and bsin4x and ce±3y
6. [A1] Obtain correct answer in any form e.g. -⅓e⁻³ʸ = ½x - ⅛ sin 4x - ⅓
7. [A1] Substitute x = ½ and obtain y = 0.175 or -ln(⅓(1 + sin 2)) OE ISW