Solving a Verhulst-Pearl model and finding point of inflection

[brandy]

Homework Statement

find p(T), then find the inflection point on the graph.

Homework Equations

dp/dt=K*P(1-P/M)
K is the growth factor
P=population
t=time
(1-p/m)=correction factor
M=maximum

The Attempt at a Solution

I integrated the given equation and got
ln (p(m-p)+c=kt c=unknown constant
p(m-p)=e^(kt+c)
p(m-p)=Ae^kt A=e^c as c=unknown constant therefore A also equals unknown constant

i said that when t=0, p=Po
and so i got Po(m-Po)=A

i fiddled around for a while and got:
p=m-(Po*(m-Po)*e^(kt))/p
but as u can see there is still a p on both sides :(
i thought about making it equal 0 and using the quadratic equation but it will take a lot of work and i don't know if it will work.


my question is what should i do? have i made a mistake somewhere?
and also, to find the point of inflection would you just double derive the p(t)?
or is there another way to do it, ie useing the dp/dt function given.

any help will be very VERY appreciated.

 

[brandy]

dw i got it