I am trying to derive an equation of motion for a simple electrostatic potential well.(adsbygoogle = window.adsbygoogle || []).push({});

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 kq_{1}q_{2}/x^{2}

Which should yield the following second order differential equation of motion

d^{2}x/dt^{2}=c/x^{2}

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

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# Looking for an analytical solution for a quadratic attractive potential well

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