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Non-dimensional differential equation

  1. Apr 20, 2014 #1

    wel

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    Gold Member

    A body of constant mass is thrown vertically upwards from the ground. It can be shown that the appropriate non-dimensional differential equation for the height [itex]x(t;u)[/itex], reached at time [itex]t\geq0[/itex] is given by
    \begin{equation} \frac{d^2x}{dt^2} = -1-\mu (\frac{dx}{dt})
    \end{equation}
    with corresponding initial conditions [itex]x(0)=0, \frac{dx}{dt}(0) =1[/itex], and where [itex]0<\mu<<1.[/itex]
    Deduce that the (non-dimensional) height at the highest point (where [itex]\frac{dx}{dt} =0[/itex]) is given by
    \begin{equation} h(\mu)= \frac{1}{\mu}- \frac{1}{\mu^2} log_e(1+\mu) \end{equation}

    =>
    It really hard for me to start

    I was thinking do integration twice by doing the separation of variable:
    \begin{equation} \frac{d^2x}{dt^2} = -1-\mu (\frac{dx}{dt})
    \end{equation}
    I got the general solution of \begin{equation}x(t)= \frac{log(t\mu +1) -t \mu}{\mu^2}\end{equation}

    after that I do not know how to get the answer.
    Please help me.
     
  2. jcsd
  3. Apr 20, 2014 #2

    LCKurtz

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    How did you get that solution? You have a constant coefficient second order DE. You wouldn't expect a logarithm in the solution.
     
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