ODEs

**housefire** · April 28th 2008, 08:52 AM

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.

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

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..

**icemanfan** · April 28th 2008, 08:56 AM

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.

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

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.

**housefire** · April 28th 2008, 09:15 AM

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..

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

where

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

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

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

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

**icemanfan** · April 28th 2008, 09:24 AM

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..

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

where

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

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

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

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 $v = \frac{dz}{dt}$ , you can use the chain rule to set the equation to $\frac{dv}{dt} = -g - \mu{v^2}$ .

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

**housefire** · April 28th 2008, 10:45 AM

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

**housefire** · April 28th 2008, 12:34 PM

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..

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