Looking for an analytical solution for a quadratic attractive potential well

[parsec]

I am trying to derive an equation of motion for a simple electrostatic potential well.

Imagine a scenario where an electron (or other charged particle) is released from an arbitrary distance from a fixed (unperturbable) attractive charge (say a proton fixed in space).

In 1 dimension, the force on the particle should be kq₁q₂/x²

Which should yield the following second order differential equation of motion

d²x/dt²=c/x²

or x''-x^-2=0

I can't seem to find an analytical solution to this equation. I'm told that due to the singularity at x=0 it will have a transcendental solution?

thanks

 

[phyzguy]

The trick to solving equations like this one, where the independent variable t is not present, is to write
x'' = x' dx'/dx. Then you have a first order equation involving x' and x. Collect terms in x' and x, then integrate both sides. Then solve for x', and integrate a second time. This will give you an analytic solution for this equation, although it will probably be more complicated than you like.