Find all the functions in $\mathbb{R}$, satisfying the given equation $(x+y)(f(x)-f(y)) = (x-y)f(x+y)$ for all $x,y$ in $\mathbb{R}$. Find all the functions in $\mathbb{R}$, satisfying the given equation 
$(x+y)(f(x)-f(y)) = (x-y)f(x+y)$ for all $x,y$ in  $\mathbb{R}$.
I tried to find something like a pattern that would become a constant. If I simplify, I get 
    $xf(x)-xf(x+y)-xf(y)=yf(y)-yf(x+y)-yf(x)$
What should I do now? Is this going to help? Thank you.  

The equation from Cesaro requires "differentiable".
A class of solutions is f(x) = ax
