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Integrating complicated functions of time

  1. Feb 7, 2012 #1
    1. The problem statement, all variables and given/known data

    m' = G O^1/2 m^(-1/2)

    where m' = dm/dt. Find m by integrating this expression wrt to time (indefinite).


    2. Relevant equations

    We have G = m'/m

    and O = constant/G^2

    and m is a function of time m(t)

    3. The attempt at a solution

    I know that integral of m'/m = ln m, so that means I can integrate the G part. I have tried to integrate O by taking the constant outside the integral, which leaves an integral of [1/(G^2)]^(1/2) or just [G^(-2)]^(1/2) or [(m'/m)^(-2)]^(1/2) I am not sure what form is easiest to work with, and have no idea how to start with this part! Could I use the substitution rule? I can't see how...

    The last part is an integral of m^(-1/2) which is 2m^(1/2).

    I also have no idea how to combine the three different parts together to give the final equation for the integral.
     
  2. jcsd
  3. Feb 7, 2012 #2

    phyzguy

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    First, put it all in terms of m and m'. If you know G in terms of m and m', and O in terms of G (which you know in terms of m and m'), then you can eliminate G and O, and get it in the form m' = f(m), where f is some function only of m (and perhaps some constants). Then you can integrate to find m in terms of t.
     
  4. Feb 7, 2012 #3
    Thanks phyzguy. I have been having a go at that, a lot of the m terms cancel out, so I am left with

    a' = a^(-0.5) {times a constant}

    Again a is some function of time. How do I integrate this wrt time t, without knowing how a and t are related? I think I am looking for the opposite of the "total differential", where you can just put a'(t) to indicate the derivative of some arbitrary function a of t. How do I do this in integration?
     
  5. Feb 7, 2012 #4

    phyzguy

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    If you have a' = a^(-n), then you can write:
    [tex] \frac{da}{dt} = a^{-n}[/tex]
    [tex]a^n da = dt[/tex]
    [tex]\int a^n da = \int dt = t + C [/tex]

    Hopefully you can do the a integral, then solve for a as a function of t.
     
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