Let y = y(x) be the solution of the differential
equation xdy = (y + x³ cosx)dx with y(π) = 0, then y( π/2 ) is equal to :

Solution

Maths sir · 4 steps

Divide by x, and the integrating factor 1/x turns the left side into d/dx(y/x).

1. Linear form. Divide both sides by x dx to get dy/dx - y/x = x² cos x. This is linear in y with P = -1/x and Q = x² cos x.
2. Integrating factor. IF = e^(∫ -1/x dx) = e⁻ˡⁿˣ = 1/x. Multiplying through by 1/x makes the left side the derivative of y/x.

   y/x = ∫ x cos x dx = x sin x + cos x + C

3. Find C. Put x = π, y = 0: 0 = π sin π + cos π + C = -1 + C, so C = 1. Hence y = x² sin x + x cos x + x.
4. So. At x = π/2: y = π²/4 × 1 + π/2 × 0 + π/2 = π²/4 + π/2, which is option (a).

The idea to keep. Always bring the equation to dy/dx + Py = Q first. Then y × IF = ∫ Q × IF dx + C, and use the given point to fix C.

Answer from the NTA final answer key. Solution by PrepWiser.