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Math Help - Non-dimensional differential equation

  1. #1
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    Non-dimensional differential equation

    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 $x(t;u)$, reached at time $t\geq0$ is given by
    \begin{equation} \frac{d^2x}{dt^2} = -1-\mu (\frac{dx}{dt})
    \end{equation}
    with corresponding initial conditions $x(0)=0, \frac{dx}{dt}(0) =1$, and where $0<\mu<<1.$
    Deduce that the (non-dimensional) height at the highest point (where $\frac{dx}{dt} =0$) 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.
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  2. #2
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    Re: Non-dimensional differential equation

    Can you show us how you got your solution.
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