Let α x=exp (x^β y^γ) be the solution of the differential equation 2 x² y d y-(1-x y²) d x=0, x > 0,y(2)=√(log ₑ 2). Then α+β-γ equals :

Solution

Maths sir · 4 steps

See 2y dy sitting there? Put t = y² and this becomes a linear equation in t.

1. Substitute. Let t = y², so dt = 2y dy. The equation becomes x² dt = (1 - xt) dx, that is, dt/dx + t/x = 1/x².
2. Integrating factor. The integrating factor is e^(∫ 1/xdx) = x for x > 0. Multiplying through gives d/dx(xt) = 1/x.

   x y² = ln x + C

3. Use y(2). At x = 2, y² = ln 2, so 2 ln 2 = ln 2 + C and C = ln 2. Then xy² = ln x + ln 2 = ln(2x), which means 2x = exp(x¹ y²).
4. So. α = 2, β = 1, γ = 2, and α + β - γ = 2 + 1 - 2 = 1, option (a).

The idea to keep. When a y dy appears next to y² terms, set t = y². The equation often turns linear in t, and the constant from the initial condition decides the final form.

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