`(dy)/(dx)=(x^(2)+x+1)/(2xy)`

(dy)/(dx)=(y^(2)-x)/(xy+y)

(dy)/(dx)=(x^(2)-y^(2))/(xy)

(dy)/(dx)=(x^(2)+xy)/(x^(2)+y^(2))

Solve (dy)/(dx)=(x^(2)+xy)/(x^(2)+y^(2))

(dy)/(dx)=(x^(2)+xy)/(x^(2)+y^(2))

(dy)/(dx)+2xy=x^(3)

(dy)/(dx)=(x^2y+y)/(xy^2+x)

x^(2)(dy)/(dx)=x^(2)+xy+y^(2)

Solution of the differentiable equation x=1+xy((dy)/(dx))+((xy)^(2))/(2!)((dy)/(dx))^(2)+((xy)^(3))/(3!)((dy)/(dx))^(3)+