
Help with an IVP
1) In recent years, Massachusetts has experienced a population explosion, not of people but of wild turkeys. The bird had virtually disappeared here when, in 1972, 37 turkeys were trucked over the border and released into the wild. There are now an estimated 20,000 of these creatures in Massachusetts. Assume that the Massachusetts wildturkey population increases at a rate proportional to its current size.
a)Write the initial value problem (differential equation plus initial condition) that models this situation. The differential equation should contain one unspecified constant.
b) Write the solution function for that initial value problem. In doing this, you will need to determine the value of the unspecified constant.
I am puzzled as to how I need to go about solving this problem. Please, any help will be much appreciated.
M

a)
$\displaystyle P_0 = 37$
$\displaystyle \frac {dP}{dt} = kP$
when t = 35, P = 20000
Now you can have a shot at b)

Does this mean that the solution function involves dP/dt=kP being implied that P(t)=Ae^kt?
P(0)=37 means that A=37: P(t)=37e^kt
P(35)=20000 = 37e^k(35)
20000=37e^35k
20000/37=e^35k
ln(20000/37)=35k
k=(1/35)ln(20000/37) = .1797877
P(t)=37e^.1797877t
Is this correct?

Looks good to me. Well done