I was solving a much larger problem and hence reduced it to a functional equation which looks a little strange but also seems easy to solve.
Question : Find all differentiable functions f defined on entire reals and differentiable on reals such that is satisfies the following
$$ f(2n) -f(-2n) = f'(2n) - f'(-2n) +4n^2 $$
for natural numbers n.
I need some help to solve this out so if anyone can help it will be appreciated and helpful.