From yesterday, you know the magnitude of the field along the axis. Taking a few liberties with the question, we can assume small displacements and hence simple harmonic motion.
For small displacements, i.e. z << R, yesterdays eqn becomes</p>
E = kqz/R3
Multiply this field by e to get the force on an electron, and divide by m to get the acceleration.
Accl = keqz/mR3
Starting to look familiar? You can solve this diff eqn if you like, or you should probably know that an eqn of the form:
x double-dot = something times x
is SHM and has an angular frequency of sqrt(something)
Can you do this without assuming small displacements? I'm not convinced you'd get the same answer - the force falls off after a bit (we found the maximum point yesterday) and this would reduce the frequency (ultimately to zero!)
I think that to get the answer they have given, you need to assume small displacements, although I am happy to be proven wrong...