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Separating Variables in Integration

  1. Feb 22, 2012 #1
    Wondering whether somebody could help me with a quick integral??
    dp/dt = ap(1-(p^2/q^2))

    initial condition p(0) = 0

    I have tried separating the variables, and then taking the partial fractions where needed, however my answer does not simplify nicely and it gets into some really complicated logarithms, I was wondering if I was doing something wrong?? Or if somebody could show me a different way of solving this equation. Thank you :)
     
  2. jcsd
  3. Feb 22, 2012 #2

    phyzguy

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    If p(0) = 0 for the differential equation you have given, dp/dt also equals zero at t=0. if both p and p' are 0 at t=0, then the only solution is that p=0 for all times.
     
  4. Feb 22, 2012 #3
    Sorry I just re-checked the question and the initial condition is p(0) = q/3, I'm really sorry about that :)
     
  5. Feb 22, 2012 #4

    phyzguy

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    Well in that case it is pretty straightforward. Write:
    [tex]\frac{dp}{p(1-p^2/q^2)} = a dt[/tex] Now integrate both sides:
    [tex]\int\frac{dp}{p(1-p^2/q^2)} = at +C[/tex]

    The integral on the left side can be done by partial fractions and will give logs, as you said. Group the logs into a single log and then exponentiate both sides, and you will get a function of p on the left side = C exp(at). Use the initial condition to determine C, solve for p, and you're done.
     
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