Question 157
Find all functions $f: \mathbb{N} \rightarrow \mathbb{N}$ such that
$$
x f(y)+y f(x)=(x+y) f\left(x^{2}+y^{2}\right)
$$
for all natural numbers $x$ and $y$.
Note: recall that $\mathbb{N}$ denotes the set of natural numbers, that is, $\mathbb{N}=\{0,1,2, \ldots\}$.
$$
x f(y)+y f(x)=(x+y) f\left(x^{2}+y^{2}\right)
$$
for all natural numbers $x$ and $y$.
Note: recall that $\mathbb{N}$ denotes the set of natural numbers, that is, $\mathbb{N}=\{0,1,2, \ldots\}$.