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 Problem Statement

A particle of mass ##m## moves under the action of a potential
##V(x)=\frac{cx}{x^2+a^2}##
where ##a## and ##c## are positive constants. Find the positions of stable equilibrium and the period of the small oscillations around those points.
 Relevant Equations

The equations of the harmonic oscillator:
##\ddot{x}=\omega^2x=0##
Period: ##\tau\simeq \frac{2\pi}{\omega}##
I tried by taking the derivative of the potential to find the critic points and the I took the second derivative to find which of those points are minimum points. I found that the point is ##x= a##. I don't understand how to calculate the period, since I haven't seen anything about the harmonic oscillator.