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Homework Statement
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
Homework Equations
dP/dt = P(a-blnP)
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