# Gompertz Model

1. Mar 11, 2009

### lockedup

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$$\int$$du/u = $$\int$$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. Mar 11, 2009

### lockedup

P = eAe-t/b?

Last edited: Mar 11, 2009
3. Mar 11, 2009

### lockedup

Or how about:

P = e(Ae-bt - a)/-b

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