Welcome to MATH226.1x: Introduction to Differential Equations. This syllabus provides a general description of the course content, the schedule, the assessments and grading, and general guidelines. Please check the syllabus if you have any questions regarding the operation of this course.
- Introduction
- About the Team
- Course Outline
- Assessments and Grading
- Discussion Forum Guidelines
- FAQ
- Time Zones
- Honor Code
- Credits and Acknowledgements
Introduction to Differential Equations
Phenomena as diverse as the motion of the planets, the spread of a disease, and the oscillations of a suspension bridge are governed by differential equations. MATH226x is an introduction to the mathematical theory of ordinary differential equations. This course adopts a modern dynamical systems approach to the subject. That is, equations are analyzed using qualitative, numerical, and if possible, symbolic techniques.
In MATH226.1, we discuss biological and physical models that can be expressed as differential equations with one or two dependent variables. We discuss geometric/qualitative and numerical techniques that apply to all differential equations. When possible, we study some of the standard symbolic solution techniques such as separation of variables and the use of integrating factors. We also study the theory of existence and uniqueness of solutions, the phase line and bifurcations for first-order autonomous systems, and the phase plane for two-dimensional autonomous systems. The techniques that we develop will be used to analyze models throughout the course.
About the Team
Paul Blanchard is professor of mathematics at Boston University. He grew up in Sutton, Massachusetts, USA,
and spent three undergraduate years at Brown University. During his senior year, he decided to have an adventure and learn a
new language, so he was an occasional student at the University of Warwick in England. He received his Ph.D. from Yale
University. He has taught mathematics for more than thirty-five years, most at Boston University. His main area of
mathematical research is complex analytic dynamical systems and the related point sets---Julia sets and the Mandelbrot set.
He is a Fellow of the American Mathematical Society.
For many of the last twenty years, his efforts have focused on modernizing the traditional sophomore-level differential equations course. That effort has resulted in numerous workshops and minicourses. He has also authored five editions of Differential Equations with Robert L. Devaney and Glen R. Hall. When he becomes exhausted fixing the errors made by his two coauthors, he heads for the golf course to enjoy a different type of frustration.
Kyle Vigil is a Ph.D. candidate in the Department of Physics at Boston University. His research involves
high numerical aperture optical systems and sub-wavelength resolution microscopy. Kyle received a Master of Arts degree
in Physics from Boston University and Bachelor of Science degrees in Mathematics and Physics from Texas A&M University.
While at Boston University he has been a teaching assistant for several Physics and Mathematics courses.
Course Outline
| Module | Content |
|
Module 1: Modeling via Differential Equations
Released on Tuesday, May 16 at 12:00 Noon EDT
|
Mathematical models use mathematical formalism to study some aspect of everyday life. Models that are expressed as differential equations involve assumptions about their rates of change. The rates are used to determine the behavior of the phenomenon in the future. We will discuss a model for the motion of a skydiver and two models of population growth. |
|
Module 2: What is a Differential Equation?
Released on Tuesday, May 16 at 12:00 Noon EDT
|
The mathematical formalism associated with a differential equation: What does it mean to write a differential equation and what does it mean to solve a differential equation. We will discuss general terminology and the "No wrong answers" principle. |
|
Module 3: Separation of Variables - An Analytic Technique
Released on Tuesday, May 23 at 12:00 Noon EDT
|
We study a technique that involves the method of substitution from integral calculus and sometimes a little algebra to solve a special type of differential equation. |
|
Module 4: Slope Fields - A Geometric and Qualitative Technique
Released on Tuesday, May 23 at 12:00 Noon EDT
|
A slope field is a picture of a differential equation. Graphs of solutions are everywhere tangent to the slope field. We learn how to sketch slope fields by hand as well as with a computer. |
|
Module 5: Euler's Method - A Numerical Technique
Released on Tuesday, May 30 at 12:00 Noon EDT
|
Euler's method is the most basic of all of the numerical algorithms that are used to approximate solutions to differential equations. We derive the method and discuss how it is implemented on a computer. |
|
Module 6: Existence and Uniqueness of Solutions
Released on Tuesday, May 30 at 12:00 Noon EDT
|
The Existence Theorem tells us that differential equations have solutions. The Uniqueness Theorem tells us when we should expect just one solution to each initial-value problem. We study these two theorems in detail along with their implications. |
|
Module 7: Autonomous Equations and their Phase Lines
Released on Tuesday, June 6 at 12:00 Noon EDT
|
Autonomous equations model "self-governing" phenomena. Their rates of change depend only on the value of the dependent variable. Examples of autonomous systems include radioactive decay, population growth subject to limited resources, and the motion of a mass-spring system. We learn how a geometric object called the phase line can be used to study autonomous, first-order equations |
|
Module 8: Bifurcations
Released on Tuesday, June 6 at 12:00 Noon EDT
|
Models that use differential equations often involve parameters, for example, the mass of a skydiver. A bifurcation value for a parameter is a value that separates one type of "long-term behavior" from another. For example, the rate of fishing each year is a parameter in a model discussed in this module. A bifurcation value would be the amount of fishing that would be the dividing line between sustainable rates and rates that lead to the collapse of the fish population. |
|
Module 9: Linear Differential Equations - Introduction and Theory, The Method of
the Lucky Guess, and The Magic Function
Released on Tuesday, June 13 at 12:00 Noon EDT
|
Introduction and Theory: Linear differential equations are especially nice differential equations because we completely understand the structure of the set of their solutions. They are also used to approximate nonlinear differential equations in certain situations. In this submodule, we discuss the structure of set of solutions. The Method of the Lucky Guess: In this submodule, we derive general solutions of certain linear equations that lend themselves to a guessing technique. The Magic Function: In theory, the general solution of any linear equation can be obtained by the use of an integrating factor. In this submodule, we derive a formula for the integrating factor, and we discuss the use of this technique in practical terms. |
|
MidMOOC Exam
Released on Tuesday, June 20 at 12:00 Noon EDT
Due on Tuesday, June 27 at 12:00 Noon EDT
|
This exam will test the topics presented in Modules 1-9. The exam will be worth 25% of your overall grade. |
|
Module 10: Systems of Differential Equations
Released on Tuesday, June 20 at 12:00 Noon EDT
|
Systems of differential equations involve two or more dependent variables rather than just one as in Modules 1 - 9. As an example, we'll discuss the predator-prey system where there are two populations that interact over time. Another important example is the mass-spring system. We discuss basic terminology and a geometric approach for understanding solutions. |
|
Module 11: Vector Fields, Direction Fields, and the Phase Plane
Released on Tuesday, June 20 at 12:00 Noon EDT
|
A vector field is a picture of a first-order system of differential equations just as a slope field is a picture of a first-order differential equation (see slope fields in Module 4 above). We learn how to sketch direction fields and study what the vector field tells us about the geometry of solutions. |
|
Module 12: The Damped Harmonic Oscillator
Released on Tuesday, June 27 at 12:00 Noon EDT
|
The damped harmonic oscillator is the second-order differential equation that is often used to model phenomena that behave linearly. The mass-spring system is the classic example. Another common example is a linear circuit. We derive a guessing technique that applies to this differential equation. We also discuss the geometry of the solutions that are obtained from this guessing technique. |
|
Module 13: Analytic Methods for Special Systems
Released on Tuesday, June 27 at 12:00 Noon EDT
|
We discuss the "no wrong answers" principle as it applies to systems of differential equations, and we present an analytic technique that applies to systems that are partially decoupled. This technique will play an important role in Module 5 of MATH226.2x. |
|
Module 14: Euler's Method for Systems
Released on Tuesday, June 27 at 12:00 Noon EDT
|
Euler's method for systems of first-order equations is similar to Euler's method for first-order equations (see Module 5 above). We discuss how the method is implemented on a computer using the vector field associated to the system. |
|
Module 15: Existence and Uniqueness for Systems
Released on Tuesday, July 4 at 12:00 Noon EDT
|
As in Module 6 (see above), we study two theorems. One guarantees that initial-value problems for systems have solutions while the other gives conditions that guarantee that solutions are unique. The Uniqueness Theorem for autonomous systems in the plane has especially important geometric implications. |
|
Module 16: The SIR Model of an Epidemic
Released on Tuesday, July 4 at 12:00 Noon EDT
|
The SIR model is a classic model for the spread of a disease. We introduce this model and use geometric techniques in the phase plane to derive the concept of a threshold value. This value predicts the onset of an epidemic. |
|
Final Exam
Released on Tuesday, July 11 at 12:00 Noon EDT
|
This exam will test all topics presented in this course and will be worth 50% of your overall grade. |
|
End of Course
Tuesday, July 18 at 12:00 Noon EDT
|
The course officially ends at this time. The content will still be available after the course closes, but those seeking a Verified Certificate must achieve an overall grade of 50% by this date. |
Assessments and Grading
Each module consists of a series of videos interspaced with brief exercises designed to help you assess your understanding of the material discussed in the video. These "content check" exercises will be worth 5% of your overall grade.
At the end of each module there will be an exercise set that will provide more detailed practice with the concepts presented in the module. These exercise sets will be worth 20% of your overall grade.
There will be a midMOOC exam that will test your overall understanding of first-order differential equations. It will be released on June 20 at 12:00 Noon (EDT). To receive credit, you must submit your answers by June 27 at 12:00 Noon (EDT). This exam will be worth 25% of your overall grade.
The final exam for the course will be released on July 11 at 12:00 Noon (EDT). It will cover all of the material discussed in all sixteen modules. To receive credit, you must submit your answers by July 18 at 12:00 Noon (EDT). The final exam will be worth 50% of your overall grade.
With the exception of the midMOOC exam, the deadline for all assessments will be the end of the course, that is, July 18 at 12:00 Noon (EDT). You may delay completion of the content check exercises and exercise sets until the end of the course while still getting credit. However, we strongly recommend that you complete all exercises as you go.
Discussion Forum Guidelines
We hope that you find the discussion forum to be a useful component of this course. It is meant to be an area where the students can interact with each other, ask questions, or talk to the course staff. We greatly encourage you to use the forum on a regular basis.
We support and encourage the use of the forum to discuss or ask questions about exercises and consequently their solutions. We will not delete questions or discussions that contain hints (even fairly substantial ones) to any of the answers; however, we do ask that you do not abuse the forum. Please do not use it as a way to share answers to exercises.
As we mention in the "Getting Help" portion of the Welcome document, we believe that one of the best ways to help people learn is to answer their questions with other questions. If you would like to help another student in this course, comments such as "Have you tried the Method of the Lucky Guess with a guess that's the sum of a sine, a cosine, and an exponential?" are appreciated. Comments such as "the answer is \(y(t)=3e^{2t} + 7 \cos 5t - 3 \sin 5t\)" are not.
We ask that you do not post comments that are derogatory, defamatory, or in any way attack other students. Be courteous and show the same respect you hope to receive. Discussion forum moderators will delete posts that are rude, inappropriate, or off-topic. Commenters who repeatedly abuse this public forum will be removed from the course.
There is a feature in the discussion forums that allows you to select from two post types, Question and Discussion. The Question type is meant for specific issues with the platform or with content, and the Discussion type is meant to share ideas and start conversation. Please keep this distinction in mind when posting to the discussion forum.
FAQ
Q
: Should I email the professor or any persons involved with this course directly?
A
: No. If you feel the need to contact the course staff, please use the
Discussion Forum.
Q
: Do I need to buy any personal materials to take this course?
A
: No. You do not need to purchase textbooks or any materials to aid you in completing the course.
Q
: I've never taken an edX course before and this is confusing. What do I do?
A
: There is an edX Walkthrough in Module 0 that beginners can watch. It explains in detail how to use the edX platform.
For further information, please visit the demo edX course.
Q
: I found a mistake in the course. Where do I report it?
A
: On the Wiki page, there is a specific section for "Errata and corrections." You can go there, edit the page, and post
information concerning any errors or issues you have found. We will try to fix them as soon as possible.
Q
: How do I learn more about the mathematics discussed in Module x?
A
: Many of the modules discuss topics that can be studied in much more detail. If you find a topic
especially interesting and would like to know more, then please post a question on the discussion forum. If we know of a good
reference or resource, then we will post it on the wiki.
Time Zones
A note about time references: Time will be reported by course staff as Eastern Daylight Time, North America (EDT). Any times listed by edX, such as due dates listed on the course site, will be reported in Coordinated Universal Time (UTC). The course staff will make every effort to make times and time zones as clear as possible. There are various time zone converters on the web such as http://www.timeanddate.com/worldclock/converter.html.
Honor Code
The edX platform assumes a certain level of decorum and responsibility from those taking this course. Please review the edX Honor Code, which is reproduced below.
By enrolling in an edX course, I agree that I will:
- Complete all mid-terms and final exams with my own work and only my own work. I will not submit the work of any other person.
- Maintain only one user account and not let anyone else use my username and/or password.
- Not engage in any activity that would dishonestly improve my results, or improve or hurt the results of others.
- Not post answers to problems that are being used to assess student performance.
Unless otherwise indicated by the instructor of an edX course, learners on edX are encouraged to:
- Collaborate with others on the lecture videos, exercises, homework and labs.
- Discuss with others general concepts and materials in each course.
- Present ideas and written work to fellow edX learners or others for comment or criticism.
Credits and Acknowledgements
As with any major effort, this course would not be possible without significant contributions from many sources. We would like to extend a special thanks to the various teams who have put in uncountable hours of work to help create this course. Specifically we want to thank the following people and organizations: Romy Ruukel, Tim Brenner, Vanessa Ruano, Diana Marian, Mathieu Hemono, and Monty Kaplan for administrating this process and being responsible for every aspect of making this course; Joe Dwyer for editing the annotated slide videos that appear in this course; Kellan Reck for filming and editing the about video; Courtney Teixeira who drew the images on the title cards; Andrew Abrahamson and Adam Brilla of BU's Metropolitan College who helped us with our tablet capture in their media room; Kacie Cleary and Arti Sharma of BU's Information Services and Technology who helped us with tablet capture in Mugar Memorial Library; Daniel Shank for accuracy checking; Professor John Polking of Rice University for letting us use his programs dfield and pplane in this course; Hubert Hohn who worked with us designing and implementing DETools, software that we use when we teach differential equations; Cengage Learning for providing partial support during the development of DETools; and the Digital Learning Initiative and the Department of Mathematics and Statistics at Boston University for supporting Paul Blanchard and Patrick Cummings during the development of this course.
Patrick Cummings worked closely with Paul Blanchard from June of 2014 until September of 2015 during the initial development phase for the course and through the first runs of 226.1, .2, and .3. His assistance was exceptional. The only thing that he could not do was keep the snow from falling during the winter of 2015 while we ran 226.1.
This course would not have been possible if the National Science Foundation had not partially funded the Boston University Differential Equations Project from 1993 to 1998.
Many undergraduate and graduate students have worked on the BU Differential Equations Project over the years: Gareth Roberts, Alex Kasman, Brian Persaud, Melissa Vellela, Sam Kaplan, Bill Basener, Sebastian Marotta, Stephanie R. Jones, Adrian Vajiac, Daniel Cuzzocreo, Duff Campbell, Lee Deville, J. Doug Wright, Dan Look, Nuria Fagella, Nick Benes, Adrian Iovita, Kinya Ono, and Beverly Steinhoff.
Paul Blanchard would especially like to thank his colleagues and coauthors, Robert L. Devaney and Glen R. Hall, for many years of enjoyable collaboration on the development of materials used to teach differential equations.
Terms of Service
For further information, please review the edX Terms of Service (https://www.edx.org/edx-terms-service)