Find all functions $f$ satisfying a certain property How can we find all real-valued functions $f$ such that $f^{(n+1)}(x) = f^{(n)}(x)$? The question gives a hint which says: "If $n=2$, then we have $f(x) = ae^x+be^{-x}$, for some $a,b \in \mathbb{R}$. After proving this proceed by induction on $n$." I can see how to complete the proof assuming the hint, but how can I prove that fact?
Clarification: We are not allowed to assume that $f(x) = ae^x+be^{-x}$, but rather need to show this when $n = 2$. 

Setting $g=f^{(n)}$, we have $g'=g$, i.e. $g(x)=be^x$, where $b$ is a real constant. Thus
$$
f^{(n)}(x)=be^x,
$$
and we deduce that
$$
f(x)=be^x+\sum_{k=0}^{n-1}a_kx^k,
$$
with $a_0,\ldots,a_{n-1}\in \mathbb{R}$.
