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Math Help - Modelling with second order ODEs

  1. #1
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    Modelling with second order ODEs

    Show that for the overdamped system:

    \dfrac{d^2x}{dt^2}+3\dfrac{dx}{dt}+2x=0

    if x=0 and \dfrac{dx}{dt} \neq 0 when t=0 then x can never equal 0 again.

    This is what I've done so far.

    The characteristic equation is:

    \lambda^2+3\lambda+2=0

    (\lambda+2)(\lambda+1)=0

    \Rightarrow \lambda=-1, -2

    x=Ae^{-2t}+Be^{-t}

    \dfrac{dx}{dt}=-2Ae^{-2t}-Be^{-t}

    When t=0, x=0

    A+B=0

    When t=0, \dfrac{dx}{dt}\neq 0

    -2A-B\neq 0

    Not too sure if I'm doing it right, or what to do next?
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  2. #2
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    What you've done so far is fine. You've also shown that since A+B=0 then B = -A.

    This gives x = Ae^{-2t} - Ae^{-t}

    x = Ae^{-t}(e^{-t} - 1).


    IF x = 0 then 0 = Ae^{-t}(e^{-t} - 1)

    Ae^{-t} = 0 or e^{-t} - 1 = 0

    e^{-t} = 0 or e^{-t} = 1

    The first case is impossible. So the ONLY time that this can happen is t = 0.
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