I found the following functional equation: Find all functions $f : \Bbb R \rightarrow \Bbb R $ such that:
$xf(x) - yf(y) = (x - y)f(x + y) $ for all $x, y \in \mathbb R $
Could you please help me? I think I proved that if $f(0) = 0$ then for each $x \in \Bbb Q$ $f(kx) = kf(x)$, but I don't know how to continue. Thanks in advance.