The solution of the differential equation (x + 2y^3 ) dy dx - y = 0 is

Step-by-step solution

The given differential equation is (x + 2y^3) dy dx - y = 0 Rearranging the terms, we get: dx dy = x + 2y^3 y dx dy - 1 y x = 2y^2 This is a linear differential equation of the form dx dy + P(y)x = Q(y) , where P(y) = - 1 y and Q(y) = 2y^2 . The integrating factor (IF) is: IF = e^ - 1 y dy = e^ - y = 1 y Multiplying the differential equation by the integrating factor and integrating, the solution is given by: x (IF) = Q(y) (IF) dy + c x ( 1 y ) = 2y^2 ( 1 y ) dy + c x y = 2y dy + c x y = y^2 + c x = y(y^2 + c) Answ