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Math Help - ODEs

  1. #1
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    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.

    \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..
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  2. #2
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    Quote Originally Posted by housefire View Post
    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.
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  3. #3
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    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..

    <br />
v \frac{dv}{dz} = -g - \mu v^2

    where

     \mu = \frac{g}{a}

    <br />
v = \frac{dz}{dt} <br />

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


    I'm sure it's probably ridiculously simple and I'm overcomplicating everything.
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  4. #4
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    Quote Originally Posted by housefire View Post
    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..

    <br />
v \frac{dv}{dz} = -g - \mu v^2

    where

     \mu = \frac{g}{a}

    <br />
v = \frac{dz}{dt} <br />

    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.
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  5. #5
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    I've tried that before and then couldn't figure out how to integrate it..
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  6. #6
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    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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