For a twice differential function $f(x)$, find all $f$ such that $f(B_t)$ is a martingale, where $B_t$ is Brownian Motion.

Question

I used Ito's lemma and came up with:

$\require{enclose} \enclose{horizontalstrike} {f_t = \dfrac{f_{xx}}{2}}$

where $f = f(B_t)$

How do I proceed further with this differential equation? Can it be solved without losing generality?

Edit: I realized that $f_t = 0$ since $f$ is independent of $B_t$. So the equation boils down to

$\dfrac {\partial d^2}{\partial x^2}f(B_t) = 0$

How does one solve this kind of differential equation. Note: Here the 'x' implies derivative w.r.t the $B_t$ variable.