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Looking for an analytical solution for a quadratic attractive potential well

  1. May 11, 2010 #1
    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


    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?

  2. jcsd
  3. May 11, 2010 #2


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    Science Advisor

    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.
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