Let y=v(x)x, then y'=v+xdv/dx and we have:
vx²(v+xdv/dx)=x²+v²x²,
v(v+xdv/dx)=1+v², assuming x≠0.
So v²+vxdv/dx=1+v², vxdv/dx=1, vdv=dx/x which integrates to v²/2=ln|x|+c, where c is the constant of integration.
Replacing v=y/x, we get (y/x)²=ln(x²)+2c, y²=x²ln(x²)+Cx² where C=2c (replacing one constant with another).