Determine all functions $f:\mathbb{R}\to \mathbb{R}$ satisfying the equation $f(a+x)-f(a-x)=4ax$, for all $a,x\in \mathbb{R}$, where any real value is available.
I came to the fact that $f(a)=0$, but I still do not know how to get the result to remove $f(x)$. Therefore, it should be confirmed that I can help here the scope of this task.
Thank you very much in advance.
Define $ g : \mathbb R \to \mathbb R $ with $ g ( x ) = f ( a + x ) - ( a + x ) ^ 2 $. Then from $$ f ( a + x ) - f ( a - x ) = 4 a x \tag 0 \label 0 $$ we get $$ \left( g ( x ) + ( a + x ) ^ 2 \right) - \left( g ( - x ) + ( a - x ) ^ 2 \right) = 4 a x \text , $$ or equivalently $ g ( - x ) = g ( x ) $; i.e. $ g $ is an even function. Conversely, for any given even function $ g : \mathbb R \to \mathbb R $, if you define $ f : \mathbb R \to \mathbb R $ with $ f ( x ) = g ( x - a ) + x ^ 2 $, then $ f $ will satisfy \eqref{0}, so those form the class of all solutions.