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Homework Help: Logistic modeling - help integrating/solving for P

  1. Apr 6, 2010 #1
    1. The problem statement, all variables and given/known data
    Sorry about the title; I accidentally hit enter instead of 'Shift'. It should read Logistic Modeling -- help integrating/solving for P

    If P(0)=2, find P(90).

    2. Relevant equations

    dP/dt=1/900P(9-P)

    3. The attempt at a solution

    my solution looks like:

    dP/(P(9-P))=1/900dt (seperable diff eq.)

    1/9(ln(P)(9-P))=1/900 t+C (partial fractions and property ln M + ln N = ln MN)

    lnP(9-P) = 1/100t+9C

    P(9-P)= Ce^(1/100t) (exponentiated; 9C=C)

    kinda stuck here (and I don't see how P ever stops growing.)
     
    Last edited: Apr 6, 2010
  2. jcsd
  3. Apr 7, 2010 #2

    lanedance

    User Avatar
    Homework Helper

    Re: Logistic

    be careful with your brackets, its pretty hard to read what is in your log, divided etc.

    but following through your method
    [tex] \frac{1}{P(9-P)} = \frac{A}{P}+\frac{b}{9-P} = \frac{9A + (b-A)P}{P(9-P)}[/tex]
    giving A = 1/9, B = 1/9

    so evaulating the integral
    [tex] \int \frac{dP}{P(9-P)} = \frac{1}{9}(\int\frac{dP}{P}+\int\frac{dP}{9-P}) =\frac{1}{9}(ln(P) - ln(9-P)) +C = \frac{1}{9}ln(\frac{P}{9-P})+C [/tex]

    maybe you missed a negative...?
     
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