Let y = y(x) be solution of the following differential equation eʸdy/dx - 2eʸ sin x + sin xcos ²x = 0,y( π/2 ) = 0 If y(0) = log ₑ(α + β e⁻²), then 4(α + β ) is equal to ______________.

Solution

Maths sir · 5 steps

Put t = eʸ and the equation becomes linear in t; one integrating factor finishes it.

1. Substitute. Let t = eʸ, so dt/dx = eʸ dy/dx. The equation becomes dt/dx - 2t sin x = -sin x cos² x, which is linear in t.
2. Integrating factor. The factor is e^(∫ -2 sin x dx) = e²ᶜᵒˢˣ. Multiplying through gives d/dx(t e²ᶜᵒˢˣ) = -sin x cos² x e²ᶜᵒˢˣ. Put u = cos x, du = -sin x dx, and integrate ∫ u² e²ᵘ du by parts.

   eʸ e²ᶜᵒˢˣ = e²ᶜᵒˢˣ((cos² x)/2 - (cos x)/2 + 1/4) + C

3. Find C. At x = π/2, y = 0 and cos x = 0: 1 = 1/4 + C, so C = 3/4.
4. Put x = 0. With cos 0 = 1: eʸ⁽⁰⁾e² = e²(1/2 - 1/2 + 1/4) + 3/4, so eʸ⁽⁰⁾ = 1/4 + 3/4e⁻².
5. So. α = 1/4 and β = 3/4, so 4(α + β) = 4 × 1 = 4.

The common slip: 1. Stopping at α + β = 1 gives 1. The question asks for 4(α + β), which is 4.

The idea to keep. When eʸ and eʸdy/dx both appear, substitute t = eʸ. The equation usually turns linear in t, and the integrating factor does the rest.

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