# Diff Eq prob from advanced dynamics

1. Oct 4, 2007

### AlphaRussOmega

1. The problem statement, all variables and given/known data
Diff eq z*z(double dot) + z(dot)^2 - gz = 0. The dots are derivatives with respect to time. Initial condition z(0)=z(not)

2. Relevant equations

3. The attempt at a solution
This is part of a larger advanced dynamics problem, where I am asked to get the velocity (z dot) as a function of z. I've spent the last seven trying to solve this by A) trying to seperate the variables, and I can't because I end up with

dv/dz + v/z = g/v

If I break up z(double dot), which is dv/dt into (dv/dz)(dz/dt), and I know know dz/dt is velocity and I replace z(dot) with velocity as well.

I also know that position z has to be a function of time, so I tried a trial solution of z=at^n, and found that the trial solution z = (ag/2)*t^2 where a is 1/3, solved for time and plugged into z(dot)=agt gives me a nice function of velocity, but is not the answer in the back of the book, and rightfully so since I haven't applied initial conditions anywhere. [I do have the right diff eq though, confirmed by the answer in the book]

So, my question is how the heck do I solve that diff eq by integrating? Is there a trick or something I'm missing to separate the variables?