This can be written: (x^3+3xy^2)dy/dx-y^3-3x^2y=0.
But d(x^3y)/dx=3x^2y+x^3dy/dx, so x^3dy/dx=d(x^3y)/dx-3x^2y.
and d(y^3x)/dx=3y^2xdy/dx+y^3, so 3y^2xdy/dx=d(y^3x)/dx-y^3.
Therefore, (x^3+3xy^2)dy/dx-y^3-3x^2y=0 can be written:
d(x^3y)/dx-3x^2y+d(y^3x)/dx-y^3-y^3-3x^2y=0;
2d(x^3y)/dx=2(y^3+3x^2y); d(x^3y)/dx=y^3+3x^2y; but d(x^3y)/dx=3x^2y+x^3dy/dx.
Therefore, y^3+3x^2y=3x^2y+x^3dy/dx and y^3=x^3dy/dx. The variables are now separable:
y^-3dy=x^-3dx.
Integrating:
-(y^-2)/2=-(x^-2)/2+k, where k is a constant; -1/2y^2=-1/2x^2+k, 1/2x^2-1/2y^2=k;
multiply through by 2x^2y^2: y^2-x^2=2kx^2y^2 or ax^2y^2, replacing 2k with a, another constant.
General solution: y^2-x^2=ax^2y^2