The variables x and θ satisfy the differential equation dx/dθ = (½x+1)sin²2θ, and x = 5 when θ = 0. Solve the differential equation and obtain an expression for x in terms of θ.

Mark scheme

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

1. [B1] Use correct double angle formula to express sin² 2θ in terms of cos 4θ sin² 2θ = ½(1-cos4θ)
2. [M1] Separate variables correctly and reasonable attempt at integration of at least one side Position of (x+5) or (x+1) and sin² 2θ sufficient for correct separation.
3. [B1] Obtain term 5ln(x+1) OE May see 5ln(x+5).
4. [B1] Obtain term ½(θ - ¼sin 4θ) Allow ½(±θ ± ¼cos 4θ) from ∫(±1 ± cos 4θ).
5. [M1] Use x = 5 when θ = 0 to evaluate a constant or as limits in a solution containing terms of the form ln(x+1), θ and sin 4θ OE
6. [A1] Obtain correct answer in any form E.g. 5ln(x+5)=½(θ-¼sin4θ) + 5ln10
7. [A1 FT] Obtain final answer x=10exp[1/10(θ-¼sin4θ)] -5 or equivalent with ln removed. Must remove ln from ln(x+5)=½(±θ±¼sin4θ) + a ln 10.