Integrating factor of the differential equation (1 + x^2) dy dx + xy = 1 is

Step-by-step solution

The given differential equation is (1 + x^2) dy dx + xy = 1 Dividing the entire equation by (1 + x^2) , we get: dy dx + x 1 + x^2 y = 1 1 + x^2 This is a linear differential equation of the form dy dx + P(x)y = Q(x) , where P(x) = x 1 + x^2 and Q(x) = 1 1 + x^2 The integrating factor (IF) is given by e^ P(x) dx P(x) dx = x 1 + x^2 dx Let 1 + x^2 = t , then 2x dx = dt x dx = dt 2 x 1 + x^2 dx = 1 2 dt t = 1 2 t = 1 2 (1 + x^2) = ( 1 + x^2 ) Therefore, the integrating factor is e^ ( 1 + x^2 ) = 1 + x^2 Answer: 1 + x^