<s>[INST] how to find all functions such that $f\left( x^{2} - y^{2} \right) = ( x - y )( f(x) + f(y) )$ Find all function $f:\mathbb R\to\mathbb R$ such that $f\left( x^{2} - y^{2} \right) = ( x - y )( f(x) + f(y) )$.
My try:
If $ x=y=0$ then $f(0)=0$ and 
if $x\leftarrow\frac{x+1}{2}$ and $y\leftarrow \frac{x-1}{2}$, then 
$f(x)=f\left( \frac{x+1}{2}  \right)+f\left( \frac{x-1}{2}  \right)$.
But how to find all functions?
 [/INST] We can show that $f(x)=-f(-x)$, hence $f(x)$ must be an odd function. If $f(x)$ is a polynomial function then we can write it as:$$f(x)=\sum_{i=0}^\infty a_ix^{2i+1}\tag{1}$$
We can also show that:$$f(x^2)=x\,f(x)\tag{2}$$
Using (1) we get:$$f(x^2)=\sum_{i=0}^\infty a_ix^{4i+2}=a_0x^2+a_1x^6+a_2x^{10}+\cdots\tag{3}$$
and:$$x\,f(x)=\sum_{i=0}^\infty a_ix^{2i+2}=a_0x^2+a_1x^4+a_2x^6+\cdots\tag{4}$$
Using (2) we know that (3) and (4) must be equivalent which means $a_i=0$ for all $i\gt0$. Hence:$$f(x)=a_0x$$
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