Let y = y(x) be the solution of the differential equation
xdy - ydx = √(x² - y²) dx, x ≥ 1, with y(1) = 0. If the area bounded by the line x = 1, x = e π , y = 0 and y = y(x) is αe 2π + β, then the value of 10(α + β) is equal to __________.

Solution

Maths sir · 4 steps

Divide by x² and the left side becomes d(y/x); after that it is one clean separation.

1. Spot the derivative. Divide both sides by x². The left side (x dy - y dx)/x² is exactly d(y/x). Put v = y/x, and since x ≥ 1 the right side becomes √(1 - v²)/x dx.
2. Separate and integrate. dv/√(1 - v²) = dx/x gives sin⁻¹ v = ln x + C. At x = 1, y = 0, so v = 0 and C = 0. Hence y = x sin(ln x).
3. Set up the area. For 1 ≤ x ≤ e^π, ln x runs from 0 to π, so sin(ln x) ≥ 0 and the curve stays above y = 0. Put x = eᵗ, dx = eᵗ dt. Then x sin(ln x) dx = e²ᵗ sin t dt with t from 0 to π.

   ∫₀^π e²ᵗ sin t dt = [e²ᵗ(2 sin t - cos t)/5]₀^π = e^(2π)/5 - (-1/5) = 1/5e^(2π) + 1/5

4. So. α = 1/5 and β = 1/5, so 10(α + β) = 10 × 2/5 = 4.

The common slip: 0. Taking the lower-limit value as +1/5 instead of -1/5 makes β = -1/5, which gives 0, not 4. At t = 0 the bracket is 2(0) - 1 = -1.

The idea to keep. Whenever you see x dy - y dx, think x² d(y/x). And for ∫ eᵃᵗ sin bt dt, use eᵃᵗ(a sin bt - b cos bt)/(a² + b²) directly.

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