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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 P

_{0}> e and 0 < P

_{0}< 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 P

_{0}> e

^{-1}and 0 < P

_{0}< e

^{-1}

**c) Find an explicit solution of the Gompertz DE subject to P(0) = P**_{0}## 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

e

^{ln (u-b)}= e

^{t + C}

u

^{-b}= Ae

^{t}

(a - b ln P)

^{-b}= Ae

^{t}

So how do I solve for P? :P Or, am I even close to having the right answer? LOL