# MA102: Single-Variable Calculus II

Unit 6: Differential Equations   This final unit will introduce the relationship between the mathematical machinery we have been developing and mathematical modeling.  In practical situations, you will rarely have all of the information or data needed to represent an initial function.  You will likely only have information about how the data changes.   In this unit, you will learn how to apply what we know about functions and how they behave in order to model and interpret data.

This unit should take you 16.5 hours to complete.

☐    Subunit 6.1: 1.5 hours

☐    Subunit 6.2: 3.5 hours

☐    Subunit 6.3: 4.75 hours

☐    Lecture: 2 hours

☐    Interactive Lab: 0.5 hours

☐    Assessment: 0.25 hours

☐    Subunit 6.4: 6.75 hours

☐    Sub-subunit 6.4.1: 2.75 hours

☐    Sub-subunit 6.4.2: 0.5 hours

☐    Sub-subunit 6.4.3: 0.5 hours

☐    Sub-subunit 6.4.4: 1.5 hours

☐    Sub-subunit 6.4.5: 1.5 hours

Unit6 Learning Outcomes
Upon successful completion of this unit, the student will be able to: - Recognize a first order differential equation. - Recognize an initial value problem. - Solve a first order ODE/IVP using separation of variables. - Draw a slope field given an ODE. - Use Euler’s method to approximate solutions to basic ODE. - Apply basic solution techniques for linear, first order ODE to problems involving exponential growth and decay, logistic growth, radioactive decay, compound interest, epidemiology, and Newton’s Law of Cooling.

6.1 First-Order Differential Equations   A differential equation represents the relationship between an unknown function and its various higher-order derivatives.  The order of the relationship is defined by the highest-ordered derivative in the equation.  In this subunit, we will only study equations involving an unknown function and its first derivative.  We will leave higher-ordered differential equations for later courses.

• Reading: Lamar University: Paul Dawkins’s Paul’s Online Math Notes: Differential Equations: “Basic Concepts: Definitions” Link: Lamar University: Paul Dawkins’s Paul’s Online Math Notes: Differential Equations: “Basic Concepts: Definitions” (HTML)

Instructions: Please click the link above, and read this entire section.  This reading introduces you to the fundamental concepts of differential equations.  Key words to remember are: order, linear differential equation, initial condition, and initial value problem.

Studying this reading should take approximately 45 minutes.

• Assessment: Clinton Community College: Elizabeth Wood’s “Supplemental Notes for Calculus I: Differential Equations and Initial Value Problems” Link: Clinton Community College: Elizabeth Wood’s “Supplemental Notes for Calculus I: Differential Equations and Initial Value Problems” (PDF)

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Instructions: Click on the link above and work through examples 1-5 on the page. As in any assessment, solve the problem on your own first. Detailed solutions are given beneath each example.

Completing this assessment should take approximately 45 minutes.

Terms of Use: The linked material above has been reposted by the kind permission of Elizabeth Wood, and can be viewed in its original form here.  Please note that this material is under copyright and cannot be reproduced in any capacity without explicit permission from the copyright holder.

6.2 Separation of Variables and Initial Value Problems   Note: Separation of variables is a method for solving certain types of differential equations.  It is based on the assumption that we can “separate” our equation into two pieces: a function of the independent variable and a function of the dependent variable, with no occurrences of one in the other.

• Reading: University of Wisconsin: H. Jerome Keisler’s Elementary Calculus: Chapter 8: Exponential and Logarithmic Functions: “Section 8.6: Some Differential Equations” Link: University of Wisconsin: H. Jerome Keisler’s Elementary Calculus: Chapter 8: Exponential and Logarithmic Functions: “Section 8.6: Some Differential Equations” (PDF)

Instructions: Click on the link above, and read Section 8.6 (pages 461 through 468).The most basic differential equation is the separable equation; in this section, you will learn how to solve such equations.

Studying this reading should take approximately 1 hour.

Terms of Use: The article above is released under a Creative Commons Attribution-NonCommercial-ShareAlike License 3.0.  It is attributed to H. Jerome Kiesler and the original version can be found here.

• Lecture: YouTube: MIT: David Jerison’s “Lecture 16: Differential Equations, Separation of Variables” Link: YouTube: MIT: David Jerison’s “Lecture 16: Differential Equations, Separation of Variables” (YouTube)

Also Available in:
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Instructions: Please click on the link above, and watch the segment of this video lecture beginning at 1:50 minutes and ending at 43:20 minutes.  Note that lecture notes are available in PDF; the link is on the same page as the lecture.  Professor Jerison discusses a number of problems involving separation of variables.  His first example is good, but somewhat more complicated than later examples.

Viewing this lecture and note-taking should take approximately 45 minutes.

Terms of Use: The video above is released under a Creative Commons Attribution-Share-Alike License 3.0 (HTML).  It is attributed to MIT and the original version can be found here (Flash or MP4).

• Assessment: Clinton Community College: Elizabeth Wood’s “Supplemental Notes for Calculus II: First Order Differential Equations” Link: Clinton Community College: Elizabeth Wood’s “Supplemental Notes for Calculus II: First Order Differential Equations” (PDF)

Also Available in:

Instructions: This assessment will cover subunits 6.1-6.2.  Click on the link above and work through each of the six examples on the page.  As in any assessment, solve the problem on your own first.  Solutions are given beneath each example.

Completing this assessment should take approximately 45 minutes.

Terms of Use: The linked material above has been reposted by the kind permission of Elizabeth Wood, and can be viewed in its original form here.  Please note that this material is under copyright and cannot be reproduced in any capacity without explicit permission from the copyright holder.

• Assessment: Temple University: Gerardo Mendoza’s and Dan Reich’s Calculus on the Web: Calculus Book II: Daniel Hartenstine’s “Differential Equations” Link: Temple University: Gerardo Mendoza’s and Dan Reich’s Calculus on the Web: Calculus Book II: Daniel Hartenstine’s “Differential Equations” (HTML)

Instructions: This assessment will cover subunits 6.1-6.2.  Click on the link above.  Then, click on the “Index” button.  Scroll down to “2. Applications of Integration,” and click button 127 (Differential Equations).  Do all problems (1-10).  If at any time a problem set seems too easy for you, feel free to move on.

Completing this assessment should take approximately 1 hour.

6.3 Slope Fields & Euler's Method   Recall that antiderivatives of functions are not unique; they differ from one another by constants.  Thus, initial values are very important in determining solutions of differential equations.  Slope fields, also called direction fields, are a way to capture the behavior of a whole family of solutions to a particular differential equation with different initial conditions.  Euler’s Method is a way to approximate solutions to differential equations numerically; it is similar in flavor to Newton’s Method.

• Reading: Lamar University: Paul Dawkins’s Paul’s Online Math Notes: Differential Equations: “Basic Concepts: Direction Fields” and “First Order DEs: Euler’s Method” Link: Lamar University: Paul Dawkins’s Paul’s Online Math Notes: Differential Equations: “Basic Concepts: Direction Fields” (HTML) and “First Order DEs: Euler’s Method” (HTML)

Studying these webpages should take approximately 2 hours.

• Lecture: YouTube: MIT: Arthur Mattuck’s “Lecture 1: The Geometrical View of y’=f(x,y): Direction Fields, Integral Curves” and “Lecture 2: Euler’s Method for y’=f(x,y) and Its Generalizations” Link: YouTube: MIT: Arthur Mattuck’s “Lecture 1: The Geometrical View of y’=f(x,y): Direction Fields, Integral Curves” (YouTube) and “Lecture 2: Euler’s Method for y’=f(x,y) and Its Generalizations” (YouTube)

Also Available in:
iTunes U

Instructions: Please click on the links above, and watch both lectures.  In these videos, Professor Mattuck will explain the concept of direction fields and do several examples.  He will state several important principles to keep in mind when sketching slope fields and integral curves and will outline Euler’s Method and discuss its error.  These are the first and second lectures for a differential equations class.

Viewing these lectures and pausing to take notes should take approximately 2 hours.

• Interactive Lab: MIT’s d’Arbeloff Interactive Math Project: “Euler’s Method” Applet Link: MIT’s d’Arbeloff Interactive Math Project: “Euler’s Method” Applet (Java)

Instructions: This is an optional resource.  Click on the link above to explore Euler’s method with this applet.  If you need more guidance, click on the “Help” link in the upper right-hand corner of the applet.

You should dedicate approximately 30 minutes to exploring this resource.

• Assessment: MIT: Haynes Miller and Arthur Mattuck’s Spring 2004 Differential Equations Class: “Problem Set 1” Link: MIT: Haynes Miller and Professor Arthur Mattuck’s Spring 2004 Differential Equations Class: “Problem Set 1” (PDF)

Instructions: Click on the link above, and then find the link to the problem set, marked “Problem Set 1.”  Do the parts of problems 1 and 2 that are intended for pencil and paper, but ignore references to the “Mathlet.”  When you are finished, click on the link above again, and find the link to the PDF with solutions to problem set

Completing this assessment should take approximately 15-20 minutes.

6.4 Exponential and Logistic Growth and Applications   A kind of bacteria might have the property where, given sufficient space, every individual produces one additional individual once an hour, so the population of bacteria doubles every hour.  If the bacteria are in a petri dish, however, there are space limitations, and the bacteria may reproduce once per hour at first but taper off dramatically as the population grows close to filling the available space.  An exponential function describes the former situation, and a logistic function describes the latter situation; such functions may also describe population decreases.  The list of possible applications is long, including the decay of radioactive material, the temperature of melting ice or cooling coffee, the number of people who have heard a rumor, and the accrual of interest in a bank account.

6.4.1 Exponential and Logistic Growth   - Web Media: UC College Prep’s Calculus BC II for AP: “Applications of Integrals: Exponential, Bounded Growth and Decay” and “Applications of Integrals: Logistic Equation and Population Growth” Link:  UC College Prep’s Calculus BC II for AP: “Applications of Integrals: Exponential, Bounded Growth and Decay” (YouTube) and “Applications of Integrals: Logistic Equation and Population Growth” (YouTube)

`````` Instructions: Click on the links above and watch the interactive
lectures.  You may want to have a pencil and paper close by, as you
will be prompted to work on related problems during the lecture.

Studying these resources should take approximately 1 hour.

are attributed to University of California College Prep.
``````
• Reading: Furman University: Dan Sloughter’s Difference Equations to Differential Equations: “6.3 Models of Growth and Decay” Link: Furman University: Dan Sloughter’s Difference Equations to Differential Equations: “6.3 Models of Growth and Decay” (PDF)

Instructions: Click the link above and read this section. This reading covers the topics outlined in sub-subunits 6.4.1-6.4.4.

Completing this reading should take approximately 30 minutes.

• Assessment: Temple University: Gerardo Mendoza’s and Dan Reich’s Calculus on the Web: Calculus Book II: Matthias Beck’s “Differential Equation of Proportional Growth” Link: Temple University: Gerardo Mendoza’s and Dan Reich’s Calculus on the Web: Calculus Book II: Matthias Beck’s “Differential Equation of Proportional Growth” (HTML)

Instructions: Click on the link above.  Then, click on the “Index” button.  Scroll down to “4. Logarithms and Exponentials, Applications,” and click button 146 (Differential Equation of Proportional Growth).  Do problems 1-4.  If at any time a problem set seems too easy for you, feel free to move on.

Completing this assessment should take approximately 1 hour.

`````` Instructions: Watch these two videos. In the first, you will learn
that the half-life of a substance is the amount of time it takes for
half of the original amount of the substance to decay through
natural processes. (Think of certain radioactive substances which
are very unstable or the carbon measured in carbon-dating.) The
second video goes over an example from biology: exponential growth
of bacteria.

Watching these lectures and pausing to take notes should take
approximately 30 minutes.

are attributed to the Khan Academy.
``````

6.4.3 Compound Interest   - Lecture: Khan Academy's “Compound Interest and e (part 2)” and “Compound Interest and e (part 3)” Khan Academy's “Compound Interest and e (part 2)” (YouTube) and “Compound Interest and e (part 3)” (YouTube)

`````` Instructions: Please click on the links above and  watch Salman
“Compound Interest and e (part 2)” and “Compound Interest and e
(part 3).”  These videos derive the formula for continuously
compounded interest and apply it.

Watching these video lectures and pausing to take notes should take
approximately 30 minutes.

are attributed to the Khan Academy.
``````

6.4.4 Epidemiology   - Web Media: UC College Prep’s Calculus BC II for AP: “Applications of Integrals: Other Examples: Spread of Disease, Rumor” The Saylor Foundation does not yet have materials for this portion of the course. If you are interested in contributing your content to fill this gap or aware of a resource that could be used here, please submit it here.

``````[Submit Materials](/contribute/)
``````
• Assessment: Furman University: Dan Sloughter’s Difference Equations to Differential Equations “6.3 Models of Growth and Decay” Link: Furman University: Dan Sloughter’s Difference Equations to Differential Equations “6.3 Models of Growth and Decay” (PDF)

Instructions: This assessment covers subunits 6.4.1-6.4.4.  Click the link above and do problems 1 (a, c, e, f), 4, 6-11, 13, and 14.  When finished, click here for solutions (courtesy of the author’s blog).

6.4.5 Newton’s Law of Cooling   - Assessment: Temple University: Gerardo Mendoza’s and Dan Reich’s Calculus on the Web: Calculus Book II: Matthias Beck and Molly M. Cow’s “Population Growth” and Dan Reich’s “Newton’s Law of Cooling” Link: Temple University: Gerardo Mendoza’s and Dan Reich’s Calculus on the Web: Calculus Book II: Matthias Beck and Molly M. Cow’s “Population Growth” (HTML) and Dan Reich’s “Newton’s Law of Cooling” (HTML)

`````` Instructions: Click on the above link.  Then, click on the “Index”
button.  Scroll down to “4. Logarithms and Exponentials,
Applications,” and click button 147 (Population Growth).  Do
problems 1-4.  Next, click button 148 (Newton’s Law of Cooling), and
do problems 1-4.  If at any time a problem set seems too easy for
you, feel free to move on.

Completing these assessments should take approximately 1 hour.