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f(xy^2)+f(x^2y)=y^2f(x)+x^2f(y), f(2008) =f(-2008)

Source: Mathcenter Contest / Oly - Thai Forum 2008 R1 p2 https://artofproblemsolving.com/community/c3196914_mathcenter_contest

November 10, 2022

algebrafunctional equationfunctional

Problem Statement

Find all the functions

f:\mathbb{R}\to\mathbb{R}

which satisfy the functional equation

f(xy^2)+f(x^2y)=y^2f(x)+x^2f(y)

for every

x,y\in\mathbb{R}

and

f(2008) =f(-2008)

(nooonuii)