If a curve y=y(x) passes through the point (1, π/2) and satisfies the differential equation (7 x⁴ cot y-eˣ cosec y) (d x)/(d y)=x⁵, x ≥ 1, then at x=2, the value of cos y is :

Solution

Maths sir · 4 steps

Flip it to dy/dx, multiply by sin y, and the substitution t = cos y turns it into a linear equation.

1. Rewrite as dy/dx. Flip the equation to get dy/dx = (7x⁴cot y - eˣcosec y)/x⁵. Multiply both sides by sin y to get sin y dy/dx = (7 cos y)/x - eˣ/x⁵.
2. Substitute cos y. Put t = cos y. Then dt/dx = -sin y dy/dx, so the equation becomes dt/dx + 7/x t = eˣ/x⁵. This is linear in t, with integrating factor e^(∫ 7/x dx) = x⁷.

   t x⁷ = ∫ x² eˣ dx = eˣ(x² - 2x + 2) + C

3. Find C. At x = 1, y = π/2, so t = cos π/2 = 0. This gives 0 = e(1 - 2 + 2) + C, so C = -e.
4. So. At x = 2: 128 cos y = e²(4 - 4 + 2) - e = 2e² - e. So cos y = (2e² - e)/128, which is option (c).

The idea to keep. When a trig function of y and its derivative both appear, multiply through so you get f'(y) dy/dx. Then substitute t = f(y) and the equation usually turns linear in t.

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