How Does an Unusual Force Affect Harmonic Oscillator Behavior?

[Onias]

Homework Statement

A particle of mass m moves (in the region x>0) under a force F = -kx + c/x, where k and c are positive constants. Find the corresponding potential energy function. Determine the position of equilibrium, and the frequency of small oscillations about it.

The Attempt at a Solution

I found the potential energy function by integrating -F(x)dx, and the position of equilibrium by putting F=0. I'm having difficulty even starting the third part, I think I have to do it like one does the damped harmonic oscillator. Any help would be greatly appreciated, thanks in advance.

 

[RoyalCat]

Hehe, nice question!

Here's a big big hint:
Work out the potential energy function while setting your plane of reference at the equilibrium point. Look at the expression you get, and examine what happens when x\rightarrow x_{eq}

Differentiate the energy function to find the force, and solve for the oscillations.