Diff Eq prob from advanced dynamics

[AlphaRussOmega]

Homework Statement

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)

Homework Equations

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 separate 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?

 

[mighty2000]

Thank you for your post. The differential equation you have provided is a second-order linear differential equation with variable coefficients. Unfortunately, there is no general method for solving such equations, and it often requires advanced techniques or numerical methods to find a solution. However, here are some steps you can take to try and solve the equation:

1. Rewrite the equation in terms of the variable t instead of z. This will make it easier to integrate.

2. Try to find a substitution that will simplify the equation. This could involve using a variable transformation or assuming a particular form for z(t).

3. If possible, try to factor the equation or use separation of variables to simplify it.

4. If all else fails, you may need to use numerical methods or consult advanced resources for help in solving the equation.

I hope this helps. Best of luck with your problem!