Pirkuliyev Rovsen wrote: Find all functions $f\colon \mathbb{R}\to\mathbb{R}$ that satisfy $f(x+y)-f(x-y)=2y(3x^2+y^2)$ for all $x,y{\in}R$ Let $P(x,y)$ be the assertion $f(x+y)-f(x-y)=2y(3x^2+y^2)$ Let $a=f(0)$ $P(\frac x2,\frac x2)$ $\implies$ $\boxed{f(x)=x^3+a}$ which indeed is a solution, whatever is $a\in\mathbb R$