Let y=y(x) be the solution curve of the differential equation (x²+1) d y d x + x y = x²+1 . If the integrating factor of this differential equation is (x²+1)² and y(0)=1 , then the value of y(1) is :

Step-by-step solution

Rewrite the given differential equation in standard linear form by dividing by (x^2+1) : d y d x + x x^2+1 y = 1 The integrating factor (IF) is given by: IF = e^ x x^2+1 d x = e^ 2 (x^2+1) = (x^2+1)^ /2 We are given that the integrating factor is (x^2+1)^2 . Comparing the powers, we get: 2 = 2 = 4 Now, multiply the standard form by the integrating factor (x^2+1)^2 : (x^2+1)^2 d y d x + 4x(x^2+1) y = (x^2+1)^2 d d x [ y(x^2+1)^2 ] = x^4 + 2x^2 + 1 Integrating both sides with respect to x : y(x^2+1)^2 = (x^4 + 2x^2 +