MA 241 – Extra Credit Assignment
Fall 2011
Applications of First- and Second-Order Differential Equations
Due: Fri. Nov. 11
Objective: This assignment is designed to explore and understand how differential equations are used in many
different scientific applications. Students will choose an application involving a first- or second-order differential
equation of their choice and complete a description and analysis of the differential equation.
Project:
1. Find an application that uses a first- or second-order (linear) differential equation. These applications can
be in any math or scientific field, but cannot include the applications discussed in class (population
growth, radioactive decay, Newton’s Law of Cooling, logistic growth, mixing problems, mass on a
spring/harmonic motion, damped motion). Feel free to consult your advisor or science professors, look up
an application in a differential equations book, or you can find an application online (just record your
source). Possible websites with applications are:
 http://college.cengage.com/mathematics/larson/calculus_applied/6e/shared/appendices/appdx_c_se
c_04.pdf
 http://www.aw-bc.com/ide/
 http://gershwin.ens.fr/vdaniel/Doc-Locale/Cours-Mirrored/Methodes-Maths/white/deqn/a1/a1intro/
a1intro.html
2. Write a description of the problem and define all variables and constants
3. Choose realistic constants and use methods discussed in class to solve the differential equation (if
possible). You may consult me if you are unsure of how to solve the differential equation.
4. . Explain in a short paragraph why you chose the application.
Example: Population Growth – Logistic Growth
Problem: A model for the spread of a rumor is that the rate of spread is proportional to the product of
the fraction of the population who have heard the rumor and the fraction who have not heard the rumor. The
𝑑𝑦
differential equation that models this is
= 𝑟𝑦(1 − 𝑦)
𝑑𝑡
Where y denotes the fraction of the population who has heard the rumor and r denotes the rate at which the
𝑃
rumor spread. This is an example of the logistic model with 𝑦 = 𝑀 and 𝑟 = 𝑘𝑀. The solution of this differential
equation is,
𝑦=
𝑦0
𝑦0 + (1 − 𝑦0 )𝑒 −𝑟𝑡
(Be sure to show all the details when you solve your differential equation)
Then, assume a small town has 1000 inhabitants. At 8 AM, 80 people have heard a rumor. By noon, half the
town has heard it. We would like to know at what time 90% of the population will have heard the rumor.
After showing some work (be sure to show this work when you do this), 90% of the population will hear the
rumor by 3:36 PM.
I chose this differential equation because I am a Sociology major and really interested in how information travels
among people. I am always intrigued by how quickly rumors spread and it is fascinating to see how we can
model this networking phenomenon just with one differential equation.
Score: You can earn up to 5 extra percentage points on the final exam.