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Thread: Modelling with second order ODEs

  1. #1
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    Modelling with second order ODEs

    Show that for the overdamped system:

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

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

    This is what I've done so far.

    The characteristic equation is:

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

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

    $\displaystyle \Rightarrow \lambda=-1, -2$

    $\displaystyle x=Ae^{-2t}+Be^{-t}$

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

    When $\displaystyle t=0, x=0$

    $\displaystyle A+B=0$

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

    $\displaystyle -2A-B\neq 0$

    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 $\displaystyle A+B=0$ then $\displaystyle B = -A$.

    This gives $\displaystyle x = Ae^{-2t} - Ae^{-t}$

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


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

    $\displaystyle Ae^{-t} = 0$ or $\displaystyle e^{-t} - 1 = 0$

    $\displaystyle e^{-t} = 0$ or $\displaystyle e^{-t} = 1$

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