Quiz 2 Key
Math 320, Lecture 2 - Spring 2009
To get the most of this quiz, allow yourself no more than 20 minutes to completely answer it, and
do not use any notes or outside help. I will grade it if you hand it in during discussion on February
16 (for Monday sections), February 18 (for Wednesday sections).
1. Problem 1 (10 points)
An epidemic hits our favorite mammal to model: bunnies. As a result, the bunnies can no
longer reproduce. The death rate is proportional to the population of the remaining bunnies.
(a) Write the differential equation for the population P(t) of rabbits at time t.
(b) Assuming that P(0) = P0, solve the initial value problem.
(c) Suppose that when the epidemic hits at t=0, there are 1500 rabbits. Six months after
crisis strikes, there are only 500 rabbits remaining. How many rabbits will be alive in 12
months (t=12)?
Solution:
This problem was meant to use remind you of the logistic equation problems we did where
dP
dt = (β−δ)P, where βand δare the rate of births and of deathes, respectively. Recall
homework problem # 17 from Section 2.2.
(a) The general equation is dP
dt = (β−δ)P, where βand δare the birth and death rates,
respectively. In our problem, β= 0 and δ=kP for some positive constant k, because
there are no births and the death rate is propornal to P. So the differential equation is
dP
dt =−kP 2.
(b) We can solve this by using the separation of variables technique.
dP
dt =−kP 2
1
P2dP =−k dt
Z1
P2dP =Z(−k)dt
−
1
P=−kt +C
1
P=kt +C0
P=1
kt +C0
At time t= 0, we have P(0) = P0. We must use this information in order to solve for our
constant C0.
P(t) = 1
kt +C0
P0=1
k(0) + C0
P0=1
C0
C0=1
P0
