Let y=y(x) be the solution of the differential equation (d y)/(d x)=((tan x)+y)/(sin x(sec x-sin x tan x)), x ∈(0, π/2) satisfying the condition y(π/4)=2. Then, y(π/3) is

Solution

Maths sir · 4 steps

Clean up that ugly denominator first; it is just sin x cos x, and then the equation is linear in y.

1. Simplify the denominator. sec x - sin x tan x = (1-sin² x)/(cos x) = cos x. So the denominator is sin x cos x, and the equation becomes dy/dx - y/(sin x cos x) = (tan x)/(sin x cos x) = sec² x.
2. Integrating factor. Write 1/(sin x cos x) = (sec² x)/(tan x), so ∫ dx/(sin x cos x) = ln(tan x). The integrating factor is e⁻ˡⁿᵗᵃⁿˣ = cot x.

   y cot x = ∫ sec² x cot x dx = ∫ dx/(sin x cos x) = ln(tan x) + C

3. Find C. At x = π/4: 2 × 1 = ln 1 + C, so C = 2.
4. So. At x = π/3: y/√3 = ln√3 + 2, so y = √3(2 + ln√3), which is option (c).

The idea to keep. Before reaching for a method, simplify the trig mess. Once it is y' + P(x)y = Q(x), the integrating factor e^(∫ P dx) does the rest.

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