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

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

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

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

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

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

The solution of the differential equation x=1+xy(dy)/(dx)+(x^(2)y^(2))/(2!)((dy)/(dx))^(2)+(x^(3)y^(3))/(3!)((dy)/(dx))^(3)+...is

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

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

the solution of (dy)/(dx)(x^(2)y^(3)+xy)=1