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)#
Let
when
Write a differential equation based on the above information. Solve the equation to find a general expression for the population
1 Answer
The GS is:
# P = 800-Ae^(- kt) #
Explanation:
We are given that
We are asked to write a differential equation based on the above information. Solve the equation to find a general expression for the population
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
# 800-P = Ae^(- kt) #
So the GS is:
# P = 800-Ae^(- kt) #