So I've been reading about functional equations and how to solve them. I found a pretty interesting problem (for me) but I think I need some help, some hint. I've never worked with this kind of problems. Well, the problem is this:
Find all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ such that for all $x, y \in \mathbb{R}$ the following equation holds:
$$xf(y)+yf(x)=(x+y)f(x)f(y)$$
I've tried with some substitutions like $x=a$ and $y=-a$ for all $a \neq 0$ it follows that $f(a)=f(-a)$, so $f$ is an even function. After that, i realized that setting $x=y=a$ for $a \neq 0$ it follows that $\Biggl(f(a)-\frac{1}{2} \Biggr)^{2}=\frac{1}{4}$ and then $f(x)=1$ or $f(x)=0$ for all $x \neq 0$. I think I'm almost done but I still don't know that happens with $f(0)$.
I would appreciate any hint to solve this problem, ¿Could you recommend any book for this kind of exercises? , ¡Thanks in advance!