If y=y(x) is the solution of the differential equation (d y)/(d x)+2 y=sin (2 x), y(0)=3/4, then y(π/8) is equal to :

Solution

Maths sir · 4 steps

Multiply by e²ˣ and integrate. At x = π/8 the whole trig part vanishes.

1. Integrating factor. The equation has the form dy/dx + Py = Q with P = 2. So the integrating factor is e^(∫ 2 dx) = e²ˣ, and the left side becomes d/dx(y e²ˣ).

   y e²ˣ = ∫ e²ˣ sin 2x dx = e²ˣ(2 sin 2x - 2 cos 2x)/(2² + 2²) + C

2. Use y(0). Dividing by e²ˣ gives y = (sin 2x - cos 2x)/4 + C e⁻²ˣ. At x = 0 this is -1/4 + C = 3/4, so C = 1.
3. At pi by 8. At x = π/8, 2x = π/4, and sin π/4 - cos π/4 = 0. The trig part drops out and only e^(-2 · π/8) remains.
4. So. y(π/8) = e^(-π/4), which is option (c).

The idea to keep. For dy/dx + ay = sin bx, use ∫ eᵃˣ sin bx dx = eᵃˣ(a sin bx - b cos bx)/(a²+b²). The solution always ends with a Ce⁻ᵃˣ term.

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