Lesson 5: A Further Investization of Rate of Change
We will investigate three separable differential equations and their solutions. These equations are each used widely to model growth and decay "real-life" situations usually related to populations. Note that these differential equations are simply rate-of-change functions.
Exploration 5.1: A Further Investigation of Rate of Change: Growth and Decay Models from a Differential Equations Point of View
As you work through and present this Exploration, try to focus on being able to present your findings with lucid explanations of the mathematics involved and on using eorrect mathematics terminology throughout.
Let
L,k, and
y, be constants. We wish to consider the following three differential equations, where
y is a function of
t.
(dy)/(dr)=ky
(dy)/(dr)=k(y-y_(0))
(dy)/(dt)=ky(L-y)
A. Verify that the three functions
y=Ce^(4),y=y_(6) Ce^(4), and
y=(L)/(1 Ce^(-1L)) are solutions, respectively, to the equations 1,2 , and 3 , where
C is a constant.
B. Use what you have learned in Calculus to solve each of the three given differential (or rate-of-change) equations for
y in order to obtain the given general solution. (The method of Partial Fractions is helpful in solving Equation 3.)
Note: Each Equation is a mathematical model for describing a physical process. Equation 1 represents Simple Growth and Decay, Equation 2 is known as Newton's Law of Cooling, while Equation 3 is a general Logistic Model.
The Logistic Model is quite important in populatica modeling and has application in other branches of mathematics, such as Chaos and Dynamics. As a population model, the constant
L is called the carrying capacity of the model, and the line
y=L is a horizontal asymptote for the solution.
15C. Problems
4. The rate of change of the number of wolves
W(n) in a population is proportional to the quantity
1500- Whi, where
t>=0 is rime measured in years and
k is the constant of proportionality. Assume, as an initial condition, that when
r=0 the wolf population is 500.
a. Write the differential equation that models this situation and show all steps in solving the equation for
W(n).
b. Find
W(t) in terms of
t and
k.
c. Use the fact that
W(4)=800, to find
k.
d. Find
\lim_(n)=mW(t).
5. The rate of growth of a population
B(t) of a certain strand of bacterium is proportional to the product of
B(t) and the quantity
1200-B(t), where
t>=0 is time measured in hours and
k is the constant of proportionality.
a. Write the differential equation that models this situation and show all steps in solving the equation for
B(t).
b. Find
B(t) in terms of
t and
k if
B(0)=200
c. Use the fact that
B(5)=700 to find
k.
d. Describe the long-term trend of this population.
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