The variables x and θ satisfy the differential equation (x / tan θ) (dx / dθ) = x² + 3. It is given that x = 1 when θ = 0. Solve the differential equation, obtaining an expression for x² in terms of θ.

Mark scheme

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

1. [B1] Separate variables correctly ∫ tan θ dθ = ∫ x/(x²+3) dx Condone missing integral signs or missing dx, dθ. Can be implied by later work.
2. [B1] Obtain term -ln(cos θ) Or equivalent e.g. ln(sec θ).
3. [M1] Obtain term of the form a ln(x²+3)
4. [A1] Obtain term (1/2)ln(x²+3)
5. [M1] Use x = 1, θ = 0 to evaluate a constant or as limits in a solution containing terms of the form a ln(x²+3) and b ln(cosθ) If they have rearranged then the constant must be of the correct form.
6. [A1] Obtain correct answer in any form (1/2)ln(x²+3) = -lncosθ+ln2.
7. [A1] Obtain final answer x² = 4/cos²θ - 3 Or equivalent e.g. x² = 4sec²x-3. lns removed.