1. ## ODEs

Okay, I have a question about motion with ODEs and I've gone round so many times and tried it in so many ways that I don't have a clue if I'm on the right line at all any more. I've ended up with this equation, where z is a function of time, t, with the rest of the unknowns as various constants. I'm trying to solve it for z but I'm having a mental block now so I'm asking for help.

Okay, I have a question about motion with ODEs and I've gone round so many times and tried it in so many ways that I don't have a clue if I'm on the right line at all any more. I've ended up with this equation, where z is a function of time, t, with the rest of the unknowns as various constants. I'm trying to solve it for z but I'm having a mental block now so I'm asking for help.

$\displaystyle \ln|z|+\frac{z}{a} = \frac{b}{g}+1$

Of course, there's a good chance I've screwed up the math before getting to this so it might not even work, in which case I may have to go and bang my head against a wall a few times..

2. Originally Posted by housefire
Okay, I have a question about motion with ODEs and I've gone round so many times and tried it in so many ways that I don't have a clue if I'm on the right line at all any more. I've ended up with this equation, where z is a function of time, t, with the rest of the unknowns as various constants. I'm trying to solve it for z but I'm having a mental block now so I'm asking for help.

Okay, I have a question about motion with ODEs and I've gone round so many times and tried it in so many ways that I don't have a clue if I'm on the right line at all any more. I've ended up with this equation, where z is a function of time, t, with the rest of the unknowns as various constants. I'm trying to solve it for z but I'm having a mental block now so I'm asking for help.

$\displaystyle \ln|z|+\frac{z}{a} = \frac{b}{g}+1$

Of course, there's a good chance I've screwed up the math before getting to this so it might not even work, in which case I may have to go and bang my head against a wall a few times..
It might help to know the differential equation you started with.

3. Okay, I was going to post it then didn't think it mattered cuz I just want to know how to solve that equation, but if you want it..

$\displaystyle v \frac{dv}{dz} = -g - \mu v^2$

where

$\displaystyle \mu = \frac{g}{a}$

$\displaystyle v = \frac{dz}{dt}$

$\displaystyle z(t=0)=0$
$\displaystyle v(t=0)=b$

I'm sure it's probably ridiculously simple and I'm overcomplicating everything.

4. Originally Posted by housefire
Okay, I was going to post it then didn't think it mattered cuz I just want to know how to solve that equation, but if you want it..

$\displaystyle v \frac{dv}{dz} = -g - \mu v^2$

where

$\displaystyle \mu = \frac{g}{a}$

$\displaystyle v = \frac{dz}{dt}$

$\displaystyle z(t=0)=0$
$\displaystyle v(t=0)=b$

I'm sure it's probably ridiculously simple and I'm overcomplicating everything.
Well, I think that to solve this equation it's easier to think in terms of v than in terms of z. Since $\displaystyle v = \frac{dz}{dt}$, you can use the chain rule to set the equation to $\displaystyle \frac{dv}{dt} = -g - \mu{v^2}$.

Then rearrange the equation to $\displaystyle \frac{dv}{-g - \mu{v^2}} = dt$ and integrate.

5. I've tried that before and then couldn't figure out how to integrate it..

6. In fact, scrap that. I've integrated it but it's not really helpful because it has a trig function in it and I need to find z in terms of a, b and g. I'm sorry to be a pain but can anyone help at all..