Question Number 55358 by gunawan last updated on 22/Feb/19
$$\mathrm{Determine}\:\mathrm{all}\:\mathrm{functions}\:{f}\::\:\mathbb{N}\:\rightarrow\:\mathbb{N}\: \\ $$$$\mathrm{satisfying} \\ $$$${xf}\left({y}\right)+{yf}\left({x}\right)=\left({x}+{y}\right){f}\left({x}^{\mathrm{2}} +{y}^{\mathrm{2}} \right) \\ $$$$\mathrm{for}\:\mathrm{all}\:\mathrm{positive}\:\mathrm{integers}\:{x}\:\mathrm{and}\:{y} \\ $$