If the solution of the differential equation (1 + x^3) dy dx + 6x^2y = 1 + x^2 is y = 1 (1 + x^3)^s [x + x^p p + x^q q + x^r r + c ] , then the LCM of p, q, r and s is...

Step-by-step solution

The given differential equation is (1 + x^3) dy dx + 6x^2y = 1 + x^2 . Dividing by (1 + x^3) , we get a linear differential equation of the form dy dx + P(x)y = Q(x) : dy dx + 6x^2 1 + x^3 y = 1 + x^2 1 + x^3 The integrating factor (IF) is given by: IF = e^ P(x) dx = e^ 6x^2 1 + x^3 dx Let 1 + x^3 = t , then 3x^2 dx = dt , which gives 6x^2 dx = 2 dt . 6x^2 1 + x^3 dx = 2 t dt = 2 t = (t^2) = ((1 + x^3)^2) Thus, IF = e^ ((1 + x^3)^2) = (1 + x^3)^2 . Multiplying the differential equation by the IF, we get: d dx [ y (