# Non-dimensional differential equation

1. Apr 20, 2014

### wel

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
\frac{d^2x}{dt^2} = -1-\mu (\frac{dx}{dt})

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
$$h(\mu)= \frac{1}{\mu}- \frac{1}{\mu^2} log_e(1+\mu)$$

=>
It really hard for me to start

I was thinking do integration twice by doing the separation of variable:
\frac{d^2x}{dt^2} = -1-\mu (\frac{dx}{dt})

I got the general solution of $$x(t)= \frac{log(t\mu +1) -t \mu}{\mu^2}$$

after that I do not know how to get the answer.