Determine all functions $f: \mathbb{R} \to \mathbb{R}$ satisfying: $$xf(y)-yf(x)=f(\frac yx)$$ for all $x, y \in \mathbb{R}; x\neq 0$.
I found that $f(1)=0$ by plugging $x=y$. Another thing I found is that by putting $y=1$, we get the function $f(x)=-f(\frac 1x)$. Now I need help solving this function.