MATH 226: Differential Equations
Course Description
Fall Term 2022
Generic
Catalog Description: This course provides
an introduction into ordinary differential equations (ODEs) with an emphasis on
linear and nonlinear systems
using analytical, qualitative, and numerical techniques. Topics will include
separation of variables, integrating factors, eigenvalue method, linearization,
bifurcation theory, and numerous applications. (MATH 0200 or by waiver) 3 hrs. lect./disc. DED
More Specific Description for Fall 2022: This course provides a modern introduction to ordinary differential equations. We will emphasize analytic, geometric and numerical approaches to solving differential equations. We will also consider how differential equations can be used as powerful tools in modeling real world phenomena.
There will be an initial focus on linear differential equations. We will use extensively the tools of linear algebra to study such equations. After discussing first order equations and systems of first order equations, we will see how higher order equations can be analyzed.
We will then turn our attention to nonlinear differential equations, with an examination of autonomous systems and questions of stability. This unit will conclude with an introduction to the exciting new area of chaos theory. We will finish the course, as time permits, with an examination of some numerical methods to approximate the solutions of differential equations.
We will explore applications
throughout by examples, problems, and projects.
Learning Goals: Students will learn aspects of
• Methods of analysis and theory for solving differential equations
analytically, as well as how to describe properties of solutions using
theoretical concepts.
• Numerical and quantitative analysis. Students will be
able to identify when and how to implement numerical methods to solve
differential equations using Maple or MATLAB.
• Graphical and qualitative
analysis/representation of differential equations and solutions.
Students will learn to create and interpret (by hand and by computer) direction
fields, phase lines, phase planes, and plots of analytical and numerical
solutions.
• Effective communication in
mathematics by receiving regular feedback for written solutions,
participating in group activities, and completing projects.
Instructor: Michael Olinick,
Office: Warner 202, Phone: 443-5559. Home telephone: 388-4290; email: molinick@middlebury.edu. Usual Office Hours:
Monday, Wednesday and Friday from 9 to 10 AM and 12:15 to 1:45 PM. I would be happy to make an appointment to
see you at other mutually convenient
times. We may have to schedule some meetings on Zoom.
Meeting Times: Monday, Wednesday and Friday, 11:15 AM - 12:05 PM in Warner 101.
Course
Website: http:// f22.middlebury.edu/MATH0226A
or follow the link from my personal webpage http://www.middlebury.edu/~molinick.
Computer
Algebra System: The
development of powerful computer algebra systems for personal computers has
revolutionized how students can investigate the behavior of many topics in
mathematics, especially differential equations. We will emphasize the use of MATLAB
and Maple, but you can also employ Mathematica
or other software.
Prerequisites: MATH 200 (Linear
Algebra)
Textbook:
James R. Brannan and William E. Boyce, Elementary
Differential Equations: An Introduction to Modern Methods and Applications, Third Edition, Wiley: 2015. The text
is available in several different formats: hardcover, loose leaf, or eTextbook. You can purchase new or used hardcover versions through
the College Store; other versions are available from Amazon or other online
vendors. A copy is also on reserve at Davis Family Library.
Your daily assignments will
include a few pages of reading in the text. Be certain to read the book
carefully (with pencil and paper, or occasionally MATLAB or Maple, close by!) Complete the relevant
reading before coming to class and before tackling the exercises.
Requirements:
There will be two midterm examinations and a final examination in addition to
required homework assignments. The midterm examinations will be given in the evening
to eliminate time pressure. Tentative dates for these tests are:
Monday, October 10
Wednesday, November 16
The College's
Scheduling Officer has set 7 – 10PM on Thursday,
December 15 as the time and date for
the final exam for our course.
Homework: Mathematics
is not a spectator sport! You must be a participant. The only effective way to
learn mathematics is to do mathematics. We may occasionally assign some
challenging problems which everyone may not be able to solve. You should,
however, make an honest attempt at every problem.
You may use your notes,
textbooks, calculators, and any computer software you have available (including
Maple ) to assist with the homework.
Bear in mind, however, that none of these will be permitted during examinations.
I encourage you to talk to
each other about the Practice Exercises
and Feedback Problems. The final write up
must be done alone. You should not have access to the assignment of
a colleague while writing up your own. Warning: The College deals quite
severely with cases of plagiarism, cheating, or other forms of academic dishonesty.
Review Middlebury’s policy on academic honesty at http://www.middlebury.edu/about/handbook/student_policies/Academic_Disciplinary_Policies
Homework must be done neatly
and legibly. Shoddy work will not be accepted for grading. Staple your
assignments! There will not be a great deal of partial credit given for obviously
incorrect answers. You should check your results where possible or at least
examine them to see whether they are plausible.
Practice
Problems: These problems
are designed to hone the skills we will learn in this course. You can complete
most problems with pencil and paper, while
some require software. We expect
you to complete all these problems but not all are graded or collected.
Feedback Problems: These are subsets of the practice problem
sets that will be collected and graded for feedback.
Work
Together, Write Alone: We
strongly encouraged you to work together in pairs or small groups. However, all
final drafts of feedback problems should be completed separately and in your
own words.
Take
Pride In Your Work. Feedback
problems are expected to be neat, organized, legible, and stapled. Poorly
written or messy work is not acceptable.
Submit
All Work On Time. No
late work will be accepted. Because we
will distribute solution sheets to assigned work on the day it is due, we can not accept late papers. You should start the
assignments early and work on them every day.
Important
Thought: One of the essential characteristics of college life
that distinguishes it from secondary school is the increased responsibility
placed on you for your own education. Most of what you will learn will not be
told to you by a teacher inside a classroom. Even if our model of you were an
empty vessel waiting passively to be filled with information and wisdom, there
would not be time enough in our daily meetings to present and explain it all.
We see you, more appropriately, as an active learner ready to confront
aggressively the often times subtle and difficult ideas our mathematics courses
contain. You will need to listen and to read carefully, to master concepts by
wrestling with numerous examples and problems and individual/team projects, and
frequently to ask thoughtful questions or make suggestions/conjectures about
the course material .
Grades: Grades in the course will be based on the two midterm
examinations, feedback problems, small group projects and the comprehensive final
exam. The relative weights of the various components of the course are roughly
as follows:
Examination 1: 25%
Examination 2: 25%
Three Projects: 20%
Final Examination: 30%
Help: Please see me immediately if you have any
difficulties with this course. Do not hesitate to utilize office hours. I
welcome questions of any sort, including questions on assignments not yet
handed in. In addition, I always appreciate your opinions, comments and
suggestions concerning the course.
Megan Paasche, ’24 will be the
homework grader for the course and will offer drop-in tutoring sessions, most likely
on Sunday and Wednesday evenings from 7:30 to 9:30 PM.
Students
may also obtain many different forms of assistance from the Center for
Teaching, Learning and Research (http://www.middlebury.edu/academics/resources/ctlr)
and the Disability Resources Center (http://www.middlebury.edu/office/disability-resource-center
) . I encourage you to investigate the services they offer.
Accommodations: Students who have Letters of Accommodation in this class are
encouraged to contact me as early in the semester as possible to ensure that
such accommodations are implemented in a timely fashion. For those without
Letters of Accommodation, assistance is available to eligible students through
Student Accessibility Services. Please contact Jodi Litchfield or Peter Ploegman through the Disability Resource
Center for more information: All discussions will remain confidential.
Expectations
• Be There: Attend all
lectures, arriving on time, and staying for the duration of the class period.
• Be Prepared: We expect students
to complete assigned readings prior to the class. Reading a mathematics text
requires a pencil and paper. Do not stress about understanding every detail you
read, but focus on getting a general picture of the
topics discussed, and understanding most of the examples. Completing these
readings will enhance the lecture experience for all of us.
• Be Present: Plan to
participate in lectures by both asking and answering questions, as well as by
taking part in discussions and group activities.
• Be Proactive in your
understanding. Complete assignments regularly. Ask questions as they come to
you. Attend office hours for clarification the moment you run into trouble.
• Be Respectful of yourself,
your classmates, your instructor, and our classroom. This is our shared
experience, and we are all partially responsible for ensuring a successful
semester as a productive, welcoming, and stimulating class environment.
• Be Honorable: Students are
expected to follow the Honor Code for all activities in this course.
Expectations for feedback assignments, exams, and projects will be discussed
explicitly in advance during class and students will be required to write/sign
the honor pledge on larger assignments.
A
Final Word: There
is a lot of exciting mathematical material in this course. Have fun with it!