A charged particle moves along a line joining two charged particles which are fixed at the locations and . Assume the interactions between the moving charged particle and each of the fixed charges is repulsive and obeys an inverse square law. The differential equation

for some constants and models the motion of the trajectory of the moving particle.

Problem:

1)Calculate the frequency of the small periodic oscillations near the equilibrium point.

2) Suppose that the initial condition is and and suppose . Find a formula for the velocity of the particle as it passes through the equilibrium position.

Attempt at solution:

I know that the equilibrium point is:

Also that the potential

I'm not really sure how to proceed for part 1 and for part 2, the only way I can imagine is to find an analytic solution to the given differential equation and plug in the initial conditions. I don't think that's the correct method to obtain the solution.

Edit:

I was able to solve part 2. I only need assistance with part 1 at the moment.

Any hints or suggestions would be greatly appreciated!