Find the particular solution of dy/dx + y cot x = 2 cos x, given that y = 2 when x = π/2.

1. 1 · Write what is given and what is to be found. Given: dy/dx + y cot x = 2 cos x and y(π/2) = 2. To find: the particular solution.
2. 2 · Write the standard form and the formula. It is dy/dx + Py = Q with P = cot x, Q = 2 cos x. Solution: y × IF = ∫(Q × IF) dx + C.
3. 3 · Find the integrating factor. IF = e^(∫cot x dx) = e^(log sin x) = sin x.
4. 4 · Integrate.
   • y sin x = ∫ 2 cos x sin x dx
   • Put t = sin x, dt = cos x dx: ∫ 2t dt = t² = sin² x
   • y sin x = sin² x + C
5. 5 · Use the condition. x = π/2, y = 2: 2 × 1 = 1 + C, so C = 1.
6. 6 · Check and box the answer. y sin x = sin² x + 1, that is y = sin x + cosec x. Check at x = π/2: y = 1 + 1 = 2.

Scoring tip: Write 'IF = e^(∫P dx)' on its own line with the answer. Examiners give a mark for the IF alone.