Solving the Gompertz Differential Equation

[lockedup]

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$$\int$$du/u = $$\int$$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

 

[lockedup]

P = e^Ae-t/b?

 

[lockedup]

Or how about:

P = e^{(Ae-bt - a)/-b}