MATH226.1xRerunSyllabus

8/19/2019 MATH226.1xRerunSyllabus

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Welcome to MATH226.1x: Introduction to Differential Equations. This syllabus providesa 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 theoperation of this course.

Introduction to Differential Equations

Phenomena as diverse as the motion of the planets, the spread of a disease, and theoscillations of a suspension bridge are governed by differential equations. MATH226x isan introduction to the mathematical theory of ordinary differential equations. This courseadopts a modern dynamical systems approach to the subject. That is, equations areanalyzed using qualitative, numerical, and if possible, symbolic techniques.

In MATH226.1, we discuss biological and physical models that can be expressed asdifferential equations with one or two dependent variables. We discussgeometric/qualitative and numerical techniques that apply to all differential equations.When possible, we study some of the standard symbolic solution techniques such asseparation of variables and the use of integrating factors. We also study the theory ofexistence and uniqueness of solutions, the phase line and bifurcations for first-orderautonomous systems, and the phase plane for two-dimensional autonomous systems. Thetechniques that we develop will be used to analyze models throughout the course.

About the Team

Paul Blanchard is professor of mathematics at BostonUniversity. He grew up in Sutton, Massachusetts, USA, andspent three undergraduate years at Brown University. Duringhis senior year, he decided to have an adventure and learn anew language, so he was an occasional student at theUniversity of Warwick in England. He received his Ph.D. fromYale University. He has taught mathematics for more thanthirty-five years, most at Boston University. His main area ofmathematical research is complex analytic dynamical systemsand the related point sets---Julia sets and the Mandelbrot set.He is a Fellow of the American Mathematical Society.

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For many of the last twenty years, his efforts have focused on modernizing the traditionalsophomore-level differential equations course. That effort has resulted in numerousworkshops 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 offrustration.

Kyle Vigil is a Ph.D. candidate in the Department of Physics atBoston University. His research involves high numericalaperture optical systems and sub-wavelength resolutionmicroscopy. Kyle received a Master of Arts degree in Physicsfrom Boston University and Bachelor of Science degrees inMathematics and Physics from Texas A&M University. Whileat Boston University he has been a teaching assistant forseveral Physics and Mathematics courses.

Course Outline

Module ContentModule 1: Modeling via DifferentialEquations

Released on Thursday, February 4 at 11:00AMEST

Mathematical models use mathematical formalismto study some aspect of everyday life. Models thatare expressed as differential equations involveassumptions about their rates of change. The rates

are used to determine the behavior of the phenomenon in the future. We will discuss a modelfor the motion of a skydiver and two models of

population growth.Module 2: What is a Differential Equation?

Released on Thursday, February 4 at 11:00AMEST

The mathematical formalism associated with adifferential equation: What does it mean to write adifferential equation and what does it mean to solvea differential equation. We will discuss generalterminology and the "No wrong answers" principle.

Module 3: Separation of Variables – AnAnalytic Technique

Released on Thursday, February 11 at11:00AM EST

We study a technique that involves the method ofsubstitution from integral calculus and sometimes alittle algebra to solve a special type of differential

equation.

Module 4: Slope Fields – A Geometric andQualitative Technique


A slope field is a picture of a differential equation.Graphs of solutions are everywhere tangent to theslope field. We learn how to sketch slope fields byhand as well as with a computer.

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Module 5: Euler’s Method – A NumericalTechnique


Euler's method is the most basic of all of thenumerical algorithms that are used to approximatesolutions to differential equations. We derive themethod and discuss how it is implemented on acomputer.

Module 6: Existence and Uniqueness ofSolutions


The Existence Theorem tells us that differentialequations have solutions. The Uniqueness Theoremtells us when we should expect just one solution toeach initial-value problem. We study these twotheorems in detail along with their implications.

Module 7: Autonomous Equations and theirPhase Lines


Autonomous equations model "self-governing" phenomena. Their rates of change depend only onthe value of the dependent variable. Examples ofautonomous systems include radioactive decay,

population growth subject to limited resources, andthe motion of a mass-spring system. We learn howa geometric object called the phase line can be usedto study autonomous, first-order equations

Module 8: Bifurcations


Models that use differential equations often involve parameters, for example, the mass of a skydiver. A bifurcation value for a parameter is a value thatseparates one type of "long-term behavior" fromanother. For example, the rate of fishing each yearis a parameter in a model discussed in this module.A bifurcation value would be the amount of fishingthat would be the dividing line between sustainablerates and rates that lead to the collapse of the fish

population.Module 9: Linear Differential Equations -Introduction and Theory, The Method ofthe Lucky Guess, and The Magic Function

Released on Thursday, March 3 at 11:00AMEST

Introduction and Theory: Linear differentialequations are especially nice differential equations

because we completely understand the structure ofthe set of their solutions. They are also used toapproximate nonlinear differential equations incertain situations. In this submodule, we discuss thestructure of set of solutions.The Method of the Lucky Guess: In thissubmodule, we derive general solutions of certainlinear equations that lend themselves to a guessingtechnique.The Magic Function: In theory, the general solutionof any linear equation can be obtained by the use ofan integrating factor. In this submodule, we derivea formula for the integrating factor, and we discussthe use of this technique in practical terms.

MidMOOC Exam

Released on Thursday, March 10 at 11:00AMESTDue on Thursday, March 17 at 1:00PM EDT

This exam will test the topics presented in Modules1 - 9. The exam will be worth 25% of your overallgrade.

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Module 10: Systems of DifferentialEquations


Systems of differential equations involve two ormore dependent variables rather than just one as inModules 1 - 9. As an example, we'll discuss the

predator-prey system where there are two populations that interact over time. Anotherimportant example is the mass-spring system. We

discuss basic terminology and a geometricapproach for understanding solutions.

Module 11: Vector Fields, Direction Fields,and the Phase Plane


A vector field is a picture of a first-order system ofdifferential equations just as a slope field is a

picture of a first-order differential equation (seeslope fields in Module 4 above). We learn how tosketch direction fields and study what the vectorfield tells us about the geometry of solutions.

Module 12: The Damped HarmonicOscillator

Released on Thursday, March 17 at 11:00AMEDT

The damped harmonic oscillator is the second-orderdifferential equation that is often used to model

phenomena that behave linearly. The mass-springsystem is the classic example. Another commonexample is a linear circuit. We derive a guessingtechnique that applies to this differential equation.We also discuss the geometry of the solutions thatare obtained from this guessing technique.

Module 13: Analytic Methods for SpecialSystems


Released Thursday, March 5 at 1:00 pm (EST)We discuss the “no wrong answers” principle as itapplies to systems of differential equations, and we

present an analytic technique that applies tosystems that are partially decoupled. This techniquewill play an important role in Module 5 ofMATH226.2x.

Module 14: Euler’s Method for Systems


Euler’s method for systems of first-order equations

is similar to Euler’s method for first-orderequations (see Module 5 above). We discuss howthe method is implemented on a computer using thevector field associated to the system.

Module 15: Existence and Uniqueness forSystems


As in Module 6 (see above), we study twotheorems. One guarantees that initial-value

problems for systems have solutions while the othergives conditions that guarantee that solutions areunique. The Uniqueness Theorem for autonomoussystems in the plane has especially importantgeometric implications.

Module 16: The SIR Model of an Epidemic


The SIR model is a classic model for the spread ofa disease. We introduce this model and usegeometric techniques in the phase plane to derivethe concept of a threshold value. This value predictsthe onset of an epidemic.

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Final Exam


This exam will test all topics presented in thiscourse and will be worth 50% of your overallgrade.

End of Course

Thursday, April 7 at 1:00PM EDT

The course officially ends at this time. The contentwill still be available after the course closes, butthose seeking a Verified Certificate must achieve anoverall grade of 50% by this date.

Assessments and Grading

Each module consists of a series of videos interspaced with brief exercises designed tohelp you assess your understanding of the material discussed in the video. These "contentcheck" 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 worth20% of your overall grade.

There will be a midMOOC exam that will test your overall understanding of first-orderdifferential equations. It will be released on March 10th at 11am (EST). To receive credit,you must submit your answers by March 17th at 1:00 pm (EDT). This exam will beworth 25% of your overall grade.

The final exam for the course will be released on Thursday, March 31st at 11:00 am

(EDT). It will cover all of the material discussed in all sixteen modules. To receive credit,you must submit your answers by April 7th at 1:00 pm (EDT). The final exam will beworth 50% of your overall grade.

With the exception of the midMOOC exam, the deadline for all assessments will be theend of the course, that is, April 7th at 1:00 pm (EDT). You may delay completion of thecontent check exercises and exercise sets until the end of the course while still gettingcredit. However, we strongly recommend that you complete all exercises as you go.

Discussion Forum Guidelines

We hope that you find the discussion forums to be a useful component of this course.They are meant to be an area where the students can interact with each other, askquestions, or talk to the course staff. We greatly encourage you to use these forums on aregular basis.

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To aid in this goal, 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 respectyou hope to receive. Discussion forum moderators will delete posts that are rude,inappropriate, or off-topic. We also ask that you do not post answers to the exercises onthese forums. You may discuss how to approach a problem or help other students who

may have questions, but please do not directly provide answers. Commenters whorepeatedly abuse this public forum will be removed from the course.

There is a new feature in the discussion forums that allows you to select from two posttypes, 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 startconversation. 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 involved in this course, please do sothrough 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 completingthe 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 indetail 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 .” You can go there, edit the

page, and post information concerning any errors or issues you have found. We will try tofix 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 youfind a topic especially interesting and would like to know more, then please post aquestion on the discussion forum. If we know of a good reference or resource, then wewill post it on the wiki.

Time Zones

A note about time references: Time will be reported by course staff as Eastern StandardTime, North America (EST) or Eastern Daylight Time, North America (EDT). Any timeslisted by edX, such as due dates listed on the course site, will be reported in Coordinated

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Universal Time (UTC). The course staff will make every effort to make times and timezones as clear as possible. There are various time zone converters on the web such ashttp://www.timeanddate.com/worldclock/converter.html .

We switch from Eastern Standard Time (EST) to Eastern Daylight Time (EDT) at 2 AM

on Sunday, March 13. Clocks are set forward one hour. Please make sure to take noticeof this time change, especially if your region does not adhere to Daylight Savings. Thetime zone converter will take this time change into account.

Honor Code

The edX platform assumes a certain level of decorum and responsibility from thosetaking 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 ownwork. 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 improveor 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 large contributionsfrom many sources. We would like to extend a special thanks to the various teams whohave put in uncountable hours of work to help create this course. Specifically, we want tothank the following people and organizations that have contributed a large amount ofeffort to make this course become a reality: Romy Ruukel, Tim Brenner, Vanessa Ruanofor administrating this process and being responsible for every aspect of making thiscourse; Joe Dwyer for filming and editing the welcome video, well as editing theannotated slide videos that appear in this course; Kellan Reck for filming and editing theabout video; Courtney Teixeira who drew the images on the title cards; Andrew

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Abrahamson and Adam Brilla of BU's Metropolitan College who helped us with ourtablet capture in their media room; Kacie Cleary and Arti Sharma of BU's InformationServices and Technology who helped us with tablet capture in Mugar Memorial Library;Daniel Shank for accuracy checking; Professor John Polking of Rice University forletting 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 teachdifferential equations; Cengage Learning for providing partial support during thedevelopment of DETools; and the Digital Learning Initiative and the Department ofMathematics and Statistics at Boston University for supporting Paul Blanchard andPatrick Cummings during the development of this course.

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 DifferentialEquations Project over the years: Gareth Roberts, Alex Kasman, Brian Persaud, MelissaVellela, Sam Kaplan, Bill Basener, Sebastian Marotta, Stephanie R. Jones, Adrian Vajiac,Daniel Cuzzocreo, Duff Campbell, Lee Deville, J. Doug Wright, Dan Look, NuriaFagella, 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 developmentof 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 ).

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MATH226.1xRerunSyllabus