If y=y(x), x ∈(0, π / 2) be the solution curve of the differential equation
(sin ² 2 x) (d y)/(d x)+(8 sin ² 2 x+2 sin 4 x) y=2 e⁻⁴ˣ(2 sin 2 x+cos 2 x),
with y(π / 4)=e^(-π), then y(π / 6) is equal to :

Solution

Maths sir · 5 steps

Divide by sin² 2x first, and the right side turns out to be an exact derivative.

1. Standard linear form. Divide through by sin² 2x. Since sin 4x = 2 sin 2x cos 2x, the term (2 sin 4x)/(sin² 2x) becomes 4 cot 2x, so P(x) = 8 + 4 cot 2x and Q(x) = 2e⁻⁴ˣ(2 sin 2x + cos 2x)/(sin² 2x).
2. Integrating factor. ∫ P dx = 8x + 2 ln(sin 2x), so the integrating factor is e⁸ˣ sin² 2x. Then Q × IF = 2e⁴ˣ(2 sin 2x + cos 2x).
3. Spot the derivative. Differentiating e⁴ˣ sin 2x gives 4e⁴ˣ sin 2x + 2e⁴ˣ cos 2x = 2e⁴ˣ(2 sin 2x + cos 2x), which is exactly Q × IF. So its integral is e⁴ˣ sin 2x + C.

   y e⁸ˣ sin² 2x = e⁴ˣ sin 2x + C

4. Find C. At x = π/4, sin 2x = 1 and y = e^(-π), so e^(-π)e^(2π) = e^π + C, giving C = 0. Hence y = e⁻⁴ˣ/(sin 2x).
5. So. At x = π/6, sin 2x = √3/2, so y = 2/√3e^(-2π/3), which is option (a).

The idea to keep. In a linear DE, multiply the right side by the integrating factor and check whether it is the derivative of a simple product before you start integrating by parts.

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