Solving logistic differential equation dP/dt = (1/900)P(9-P) with P(0)=2

[brusier]

Homework Statement

Sorry about the title; I accidentally hit enter instead of 'Shift'. It should read Logistic Modeling -- help integrating/solving for P

If P(0)=2, find P(90).

Homework Equations

dP/dt=1/900P(9-P)

The Attempt at a Solution

my solution looks like:

dP/(P(9-P))=1/900dt (seperable diff eq.)

1/9(ln(P)(9-P))=1/900 t+C (partial fractions and property ln M + ln N = ln MN)

lnP(9-P) = 1/100t+9C

P(9-P)= Ce^(1/100t) (exponentiated; 9C=C)

kinda stuck here (and I don't see how P ever stops growing.)

 

[lanedance]

be careful with your brackets, its pretty hard to read what is in your log, divided etc.

but following through your method
$$\frac{1}{P(9-P)} = \frac{A}{P}+\frac{b}{9-P} = \frac{9A + (b-A)P}{P(9-P)}$$
giving A = 1/9, B = 1/9

so evaulating the integral
$$\int \frac{dP}{P(9-P)} = \frac{1}{9}(\int\frac{dP}{P}+\int\frac{dP}{9-P}) =\frac{1}{9}(ln(P) - ln(9-P)) +C = \frac{1}{9}ln(\frac{P}{9-P})+C$$

maybe you missed a negative...?