What is the differential equation?

Let #P(t)# represent the number of wolves in a population at time #t# years,
when #t>=0#. The population #P# is increasing at a rate directly proportional to #800-P#, where the constant of proportionality is #k#.
Write a differential equation based on the above information. Solve the equation to find a general expression for the population #P(t)#

1 Answer
Steve M
May 25, 2018

The GS is:

# P = 800-Ae^(- kt) #

Explanation:

We are given that #P(t)# represent the number of wolves in a population at time #t# years, and #P# is increasing at a rate directly proportional to #800-P#, where the constant of proportionality is #k#.

We are asked to write a differential equation based on the above information. Solve the equation to find a general expression for the population #P(t)#

So using the description we have:

# \ \ \ \ \ (dP)/dt prop 800 - P #

# :. (dP)/dt = k(800-P)#

Which is a Separable ODE, so we can "separate the variables" to get:

# int \ 1/(800-P) \ dP = int \ k \ dt#

Which we can integrate to get:

# -ln|800-P| = kt+ C #

And we can rearrange:

# ln|800-P| = - kt- C #
# :. |800-P| = e^(- kt- C) #

And noting that #e^x gt 0 AA x in RR# we can write

# 800-P = Ae^(- kt) #

So the GS is:

# P = 800-Ae^(- kt) #

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