Let y=y(x) be the solution of the differential equation
2 cos x (d y)/(d x)=sin 2 x-4 y sin x, x ∈(0, π/2). If y(π/3)=0, then y′(π/4)+y(π/4) is equal to _________.

Solution

Maths sir · 5 steps

Divide by 2 cos x to reach standard linear form, and the integrating factor sec² x does the rest.

1. Standard linear form. Write sin 2x = 2 sin x cos x and divide the whole equation by 2 cos x. This is allowed because cos x ≠ 0 on (0, π/2). You get dy/dx + 2 tan x · y = sin x.
2. Integrating factor. The integrating factor is e^(∫ 2 tan x dx) = e²ˡⁿˢᵉᶜˣ = sec² x. Multiplying by it turns the left side into d/dx(y sec² x), and the right side becomes sin x sec² x = sec x tan x.

   y sec² x = ∫ sec x tan x dx = sec x + C ⇒ y = cos x + C cos² x

3. Use the condition. y(π/3) = 1/2 + C/4 = 0, so C = -2 and y = cos x - 2 cos² x. Differentiating gives y' = -sin x + 4 sin x cos x.
4. Substitute the point. At x = π/4, sin x = cos x = 1/√2. So y = 1/√2 - 1 and y' = -1/√2 + 2.
5. So. y'(π/4) + y(π/4) = (2 - 1/√2) + (1/√2 - 1) = 1.

The idea to keep. Always bring a linear equation to the form y' + P(x) y = Q(x) first, with the coefficient of y' equal to 1. Only then is the integrating factor e^(∫ P dx).

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