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Math Help - Rumour problem (logistic differential equations)

  1. #1
    Junior Member
    Joined
    Feb 2008
    Posts
    60

    Rumour problem (logistic differential equations) - somewhat urgent

    Hello. I'm currently stuck on a problem that I can't seem to work out. It's quite complex and difficult (for me anyway) and it would be great if you could help me out. Below is the background required:

    Consider a population N where R members know the rumour and meet one of (N - R) members who doesn't know the rumour.

    The rate of change in the population who know the rumour is hence, given by:

    \frac{dR}{dt} = k(N - R)R where k is some constant.

    ---------------------------

    Here's the question I'm stuck with:

    A town has N residents. At 8 am, a rumour begins with 2 people (spread by a logistic growth). At 9 am, 22 know the rumour. At 12 pm, half the population know the rumour. Find the population and k.

    First, I transformed it into a differential equation and proceeded to try and solve it:

    \frac{dR}{dt} = k(N - R)R

    \int \! \frac{1}{R(N - R)} \, dR = \int \! k \, dt

    \int \! \frac{1}{N}\left (\frac{1}{R} + \frac{1}{N - R}\right ) \, dR = \int \! k \, dt

    \frac{1}{N}\int \! \left (\frac{1}{R} + \frac{1}{N - R}\right ) \, dR = \int \! k \, dt

    \frac{1}{N}(ln |R| - ln |N - R|) = kt + c

    ln \left |\frac{R}{N - R}\right | = Nkt + c

    Now I'm stuck...It sounds as easy as plugging in the values where:

    t = 0
    R = 2

    t = 1
    R = 22

    t = 4
    R = 0.5N

    And solve it simultaneously. However, I can't seem to get it (did I do something wrong?)

    If you could assist me from this point on, it would definitely be appreciated.

    The answers given by the textbook is:

    k = 0.00008169
    Popluation = 29360

    Thanks.
    Last edited by sqleung; October 15th 2008 at 01:39 AM.
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  2. #2
    Super Member
    Joined
    Aug 2008
    Posts
    903
    I think it should be \ln\left(\frac{R}{N-R}\right)=nkt+nc

    but even so, I end up with the equations:

    c=1/N\ln\left(\frac{2}{N-2}\right)
    k+c=1/N\ln\left(\frac{22}{N-22}\right)
    c=-4k

    or:

    -4k=1/N\ln\left(\frac{2}{N-2}\right)
    -3k=1/N\ln\left(\frac{22}{N-22}\right)

    which looks to me to be best solved numerically. Maybe another way though.
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