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Homework Help: Gompertz Model

  1. Mar 11, 2009 #1
    1. The problem statement, all variables and given/known data
    The problem in the book:

    a) Suppose a = b = 1 in the Gompertz differential equation. Since the DE is autonomous, use the phase portrait concept of Section 2.1 to sketch representative solution curves corresponding to the cases P0 > e and 0 < P0 < e.

    b) Suppose a = 1, b = -1 in the Gompertz DE. Use a new phase portrait to sketch representative solution curves corresponding to the cases P0 > e-1 and 0 < P0 < e-1

    c) Find an explicit solution of the Gompertz DE subject to P(0) = P0


    2. Relevant equations
    dP/dt = P(a-blnP)


    3. The attempt at a solution
    I used separation of variables to get:
    dP/(P(a-blnP)) = dt

    I let u = a - blnP and du = -bdP/P which leaves me with:
    -b[tex]\int[/tex]du/u = [tex]\int[/tex]dt

    I integrate to get:
    -b ln (u) = t + C

    ln (u-b) = t + C

    eln (u-b) = et + C

    u-b = Aet

    (a - b ln P)-b = Aet

    So how do I solve for P? :P Or, am I even close to having the right answer? LOL
     
  2. jcsd
  3. Mar 11, 2009 #2
    P = eAe-t/b?
     
    Last edited: Mar 11, 2009
  4. Mar 11, 2009 #3
    Or how about:

    P = e(Ae-bt - a)/-b
     
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