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Green's Function with repulsive force

  1. Sep 30, 2015 #1
    1. The problem statement, all variables and given/known data
    Consider an object subject to a linear repulsive force, ##F = kx##. Show that the Green’s function for this object is given by:

    ##G(t-t^\prime)=\frac{1}{m\gamma}sinh(\gamma(t-t^\prime))##

    where ##\gamma=\sqrt{\frac{k}{m}}##

    2. Relevant equations
    ##sinhx=\frac{e^x+e^{-x}}{2}##

    3. The attempt at a solution
    So when I'm trying to make the Green's Function I start with:

    ##x(t)=\int_0^tG(t-t^\prime)F(t^\prime)dt^\prime##

    ##F(t^\prime)=kx## so:

    ##x(t)=\int_0^tG(t-t^\prime)(kx)dt^\prime##

    For an impulse I think ##G(t-t^\prime)=\frac{v(t-t^\prime)}{\Delta p}## where ##\Delta p = mv-0=mv##. However I don't know what to do for a repulsive force. Also I don't know how to get to the ##sinh## part. Any help would be greatly appreciated.
     
  2. jcsd
  3. Sep 30, 2015 #2

    gabbagabbahey

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    Hint: ##F=kx## tells you that ##m\ddot{x}(t) = kx(t)##...what does your equation for the Green's Function tell you ##\ddot{x}(t)## is?
     
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