Let y=y(x) be the solution curve of the differential equation (1+x^2)^2 dy dx + 2x(1+x^2)y = 1 . If _ x y(x)(1+x^2) = , then the value of y(1) is equal to

Step-by-step solution

Given differential equation: (1+x^2)^2 dy dx + 2x(1+x^2)y = 1 Dividing the entire equation by (1+x^2)^2 , we get: dy dx + 2x 1+x^2 y = 1 (1+x^2)^2 This is a linear differential equation of the form dy dx + P(x)y = Q(x) . The integrating factor (IF) is: IF = e^ 2x 1+x^2 dx = e^ (1+x^2) = 1+x^2 The general solution is given by: y (1+x^2) = 1 (1+x^2)^2 (1+x^2) dx y(1+x^2) = 1 1+x^2 dx y(1+x^2) = ⁻¹(x) + C We are given the condition _ x y(x)(1+x^2) = . Taking the limit on both sides of the solution: _ x ( ⁻¹(x) + C) =