solve the functional equation $f(x+t)-f(x-t)=4xt$ I think this question might be related with arbitrary functions, but I’m not sure. I also tried to set $t$ to different values but couldn’t get it to work.
I tried to set $t=x$ and end up with $f(2x)=f(0)+4x^2$, $f(x)=f(0)+x^2$.

Let $f(q) = q^2$ + const., that solves it directly. Is your question for a solution or for all solutions?
