A curve C: y = y(x) lies in the region x < 1 and satisfies the differential equation (x^2+1) dy - (x y + y^2 x^2+1 ) dx = 0 . If the curve passes through the point (0, 1) and y( ) = 5 , then the value of is

Step-by-step solution

The given differential equation can be rewritten as: (x^2+1) dy dx - xy = y^2 x^2+1 dy dx - x x^2+1 y = y^2 x^2+1 This is a Bernoulli differential equation. Dividing the entire equation by y^2 , we get: y⁻² dy dx - x x^2+1 y⁻¹ = 1 x^2+1 Let v = y⁻¹ . Then dv dx = -y⁻² dy dx . Substituting this in, we obtain a linear differential equation: - dv dx - x x^2+1 v = 1 x^2+1 dv dx + x x^2+1 v = - 1 x^2+1 The integrating factor (I.F.) is: I.F. = e^ x x^2+1 dx = e^ 1 2 (x^2+1) = x^2+1 Multiplying the linear equation by the