Let $f:\mathbb{R}\to \mathbb{R}$ and such for any real numbers $x,y$ we have
 $$f(xf(y)+yf(x))=yf(x)+xf(y)$$
Find $f(x)$.

I have let $x=y=0$ have
$$f(0)=2f(0)\Longrightarrow f(0)=0$$

and I guess the solution is $f(x)=x$ or $f(x)=0$, but I can't prove it.