Let y=vx, then dy/dx=v+xdv/dx. The DE becomes:
x3+v3x3-3v2x3(v+xdv/dx)=0,
1+v3-3v3=3v2xdv/dx,
xdv/dx=(1-2v3)/(3v2),
(3v2/(1-2v3))dv=dx/x, integrating:
∫(3v2/(1-2v3))dv=ln(Ax) where A is a constant.
Let u=1-2v3, then du/dv=-6v2, dv=-du/6v2,
∫(3v2/u)(-du/6v2)=ln(Ax),
-½∫du/u=-½ln(u)=ln(Ax),
ln(u)=-2ln(Ax)=ln(C/x2), where C is a constant,
u=C/x2,
1-2v3=C/x2,
1-2y3/x3=C/x2,
x3-2y3=Cx, x3-2y3-Cx=0.