Let y = y(x) satisfies the equation dy/dx - |A| = 0, for all x > 0, where A = [ y, sin x, 1; 0, - 1, 1; 2, 0, 1/x ]. If y(π ) = π + 2, then the value of y( π/2 ) is :

Solution

Maths sir · 4 steps

Open the determinant first and you get a linear equation in y with integrating factor x.

1. Expand the determinant. Expand |A| along the first row: |A| = y(-1/x - 0) - sin x (0 - 2) + 1 (0 + 2) = -y/x + 2 sin x + 2.
2. Linear form. So dy/dx + y/x = 2 sin x + 2. Here P = 1/x, so the integrating factor is e^(∫ 1/xdx) = x for x > 0.

   xy = ∫ (2x sin x + 2x) dx = -2x cos x + 2 sin x + x² + C

3. Find C. At x = π, y = π + 2: the left side is π² + 2π and the right side is 2π + 0 + π² + C. So C = 0.
4. So. At x = π/2: π/2 y = 0 + 2 + π²/4, so y = π/2 + 4/π. The answer is (a).

The idea to keep. When a determinant hides inside a differential equation, expand it first and collect the y terms. It usually turns into a linear equation y' + Py = Q with a neat integrating factor.

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