How Do I Solve This Interesting But Challenging ODE?

[gvk]

Very simple in writing, but hard to solve:
$$x*\frac{dy}{dx}=(y^2-y x^2)/(y^2+x^2)$$

 

[amcavoy]

Very simple in writing, but hard to solve:
$$x*\frac{dy}{dx}=(y^2-y x^2)/(y^2+x^2)$$

Maybe convert to polar coodinates? That x²+y² wants to be r². Maybe not though

Alex

 

[saltydog]

Very simple in writing, but hard to solve:
$$x*\frac{dy}{dx}=(y^2-y x^2)/(y^2+x^2)$$

I tend to look at it this way:

$$(y^2-yx^2)dx-(xy^2+x^3)dy=0$$

but are unable to proceed. It's not exact, can't make it exact, and can't come up with any differentials by inspection.