If y = y(x),y ∈ [ 0,π/2 ) is the solution of the differential equation sec ydy/dx - sin (x + y) - sin (x - y) = 0, with y(0) = 0, then 5y'( π/2 ) is equal to ______________.

Solution

Maths sir · 4 steps

Turn the two sines into one product, then divide by sec y; the variables separate on their own.

1. Combine the sines. Use sin(x+y) + sin(x-y) = 2 sin x cos y. The equation becomes sec y dy/dx = 2 sin x cos y, so dy/dx = 2 sin x cos² y.
2. Separate and integrate. Divide by cos² y to get sec² y dy = 2 sin x dx. Integrating gives tan y = -2 cos x + C. Putting in y(0) = 0 gives 0 = -2 + C, so C = 2.

   tan y = 2 - 2 cos x

3. Value at π/2. At x = π/2, tan y = 2, so sec² y = 1 + 4 = 5 and cos² y = 1/5. Then y'(π/2) = 2 × 1 × 1/5 = 2/5.
4. So. 5y'(π/2) = 5 × 2/5 = 2.

The common slip: 0. Writing sin(x+y) + sin(x-y) as 2 cos x sin y makes y' carry a factor of cos x, which vanishes at π/2 and gives 0. The correct identity is 2 sin x cos y.

The idea to keep. To find y' at a point, you rarely need y itself. Get y' in terms of x and y from the equation, then use the solution only to find tan y or cos y there.

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