If for the solution curve y=f(x) of the differential equation (d y)/(d x)+(tan x) y=(2+sec x)/(1+2 sec x)², x ∈(-π/2, π/2), f(π/3)=√3/10, then f(π/4) is equal to:

Solution

Maths sir · 4 steps

Multiply by sec x, and the right side turns out to be the exact derivative of (sin x)/(2+cos x).

1. Integrating factor. The equation is linear in y with P = tan x. So the integrating factor is e^(∫ tan x dx) = sec x, which is positive on (-π/2, π/2). That gives d/dx(y sec x) = (sec x (2+sec x))/(1+2 sec x)².
2. Clean the right side. Multiply its top and bottom by cos² x. It becomes (1+2 cos x)/(2+cos x)², and this is exactly the quotient-rule derivative shown below.

   d/dx((sin x)/(2+cos x)) = (cos x(2+cos x)+sin² x)/(2+cos x)² = (1+2 cos x)/(2+cos x)²

3. Find the constant. So y sec x = (sin x)/(2+cos x) + C. At x = π/3 the left side is √3/10× 2 = √3/5. The right side is (√3/2)/(5/2) + C = √3/5 + C, so C = 0 and y = (sin x cos x)/(2+cos x).
4. So. At x = π/4, y = (1/2)/(2+1/√2) = √2/2(2√2+1). Multiply the top and bottom by 2√2-1 to get √2(2√2-1)/(2 × 7) = (4-√2)/14. That is option (b).

The idea to keep. When the right side of a linear DE looks ugly, multiply it by the integrating factor and clear the secants into cosines. It is often a quotient-rule derivative you can spot directly.

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