Symmetric functional equation $ x f ( y ) + y f ( x ) = ( x + y ) f \left( x ^ 2 + y ^ 2 \right) $ on $ \mathbb N _ 0 $: how to show $ f$ is constant?

Question

Let $ \mathbb N _ 0 $ denote the non-negative integers. Find all functions $ f : \mathbb N _ 0 \to \mathbb N _ 0 $ such that $$ x f ( y ) + y f ( x ) = ( x + y ) f \left( x ^ 2 + y ^ 2 \right) \quad \forall \, x, y \in \mathbb N _ 0 $$

I got that $ f ( x ) = f \left( 2 x ^ 2 \right) $ for non zero $ x $ by setting $ x = y $. Does this show $ f $ is always constant, since the constant case does work by plugging in?

Answer 1

Hint: Plug in $y=0$ to find that $xf(x^2)=xf(0)$ for all $x$.