Let y=y(x) be the solution of the differential equation (x^2+1) dy dx + 2xy = 1 x^2+1 such that _ x x^2 y(x) = . Then y(1) is equal to

Step-by-step solution

The given differential equation can be written in the standard linear form as: dy dx + 2x x^2+1 y = 1 (x^2+1)^2 Integrating Factor (I.F.) = e^ 2x x^2+1 dx = e^ (x^2+1) = x^2+1 Multiplying the differential equation by the I.F. and integrating, we get: y(x^2+1) = 1 x^2+1 dx y(x^2+1) = ⁻¹ x + C y(x) = ⁻¹ x + C x^2+1 Given that _ x x^2 y(x) = : _ x x^2 x^2+1 ( ⁻¹ x + C) = 1 ( 2 + C ) = C = 2 Therefore, the solution is y(x) = ⁻¹ x + 2 x^2+1 Substituting x=1 : y(1) = ⁻¹(1) + 2 1^2+1 = 4 + 2 2 = 3 4 2 = 3 8 Answer: 3 8