Ordinary Differential Equations
Math E-21c - Harvard University Extension School

Lectures by: Robert Winters
email: robert@math.rwinters.com

Recitations/Zoom Office Hours by:
Kris Lokere, Jeremy Marcq, Renee Chipman

Canvas site

Lecture Notes and
Supplementary Materials

Though there is no required text for the course, several good texts may serve as useful references. They are:

Elementary Differential Equations with Boundary Value Problems, 6th Edition by Edwards & Penney (ISBN 0130339679 for 2008 hardcover edition) or an earlier edition.
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Table of Contents:

• Ch. 1 - First-Order Differential Equations

• Ch. 2 - Linear Equations of Higher Order

• Ch. 3 - Power Series methods

• Ch. 4 - Laplace Transform Methods

• Ch. 5 - Linear Systems of Differential Equations

• Ch. 6 - Numerical Methods

• Ch. 7 - Nonlinear Systems and Phenomena

• Ch. 8 - Fourier Series Methods

• Ch. 9 - Eigenvalues and Boundary Value Problems

Differential Equations & Linear Algebra by Farlow, Hall, McDill, West. This text was used for the Spring 2014 semester and is published by Pearson and has ISBN #9780131860612.
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Table of Contents:

1. First-Order Differential Equations

2. Linearity and Nonlinearity

3. Linear Algebra

4. Higher-Order Linear Differential Equations

5. Linear Transformations

6. Linear Systems of Differential Equations

7. Nonlinear Systems of Differential Equations

8. Laplace Transforms

9. Discrete Dynamical Systems

10. Control Theory and the Appendices

Differential Equations & Linear Algebra (2nd Edition) by Edwards & Penney
ISBN: 0131481460
ISBN 13: 9780131481466
Publisher: Prentice Hall Publication
Date: 2004; 784 pages
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Table of Contents:

1. First Order Differential Equations

2. Mathematical Models and Numerical Methods

3. Linear Systems and Matrices

4. Vector Spaces

5. Higher Order Linear Differential Equations

6. Eigenvalues and Eigenvectors

7. Linear Systems of Differential Equations

8. Matrix Exponential Methods

9. Nonlinear Systems and Phenomena

10. Laplace Transform Methods

11. Power Series Methods

Appendix A. Existence and Uniqueness of Solutions

Appendix B. Theory of Determinants

Earlier editions and related editions of the Edwards & Penney text may also be good reference texts.

Announcements:

Final Exams were received from everyone who were expected to take the Final Exam.
Course grades will be available from the Harvard Extension School on Tuesday, January 7th.
All exams have now been compressed to a manageable size with appropriate resolution.
There are 129 exams to grade in Math E-21a (plus the submitted HW12 extra problem), and 33 exams to grade in Math E-21c.
This will take some time, but I will try to get everything done as soon as possible.
Happy holidays and thank you to everyone who took the course! - Robert Winters

Course Meeting Times:
Lectures (Robert Winters): Wednesdays, 8:10pm-10:10pm via Zoom

Syllabus for Math E-21c (Fall 2024)    Printable syllabus (PDF, Fall 2024)     [updated July 22, 2024]

Lecture Notes and Supplementary Materials     Course Calendar (with links to notes and supplements)

This course is being offered in an online web conference format (live or on demand). All lectures will be recorded and available via the Canvas site for the course. Additional Recitations/Office Hours may be scheduled based on demand.

Optional Recitations (by Teaching Assistants Renée Chipman, Kris Lokere, Jeremy Marcq)

This course was offered for the first time in the Fall 2020 semester at the Harvard Extension School. The course has been adapted from the corresponding MIT mathematics course (18.03). Some of the topics listed in the syllabus may be modified to fit into the current course format.

This course is a study of Ordinary Differential Equations (ODE’s), including modeling physical systems. Topics will likely include:

• Solution of First-order ODE’s by Analytical, Graphical and Numerical Methods;
• Linear ODE’s, Especially Second Order with Constant Coefficients;
• Basic ideas of Linear Algebra applied to ODEs - linear spaces, span, linear independence, basis for a subspace;
• Undetermined Coefficients and Variation of Parameters;
• Sinusoidal and Exponential Signals: Oscillations, Damping, Resonance;
• Complex Numbers and Exponentials;
• Fourier Series, Periodic Solutions;
• Delta Functions, Convolution, and Laplace Transform Methods (at least the basic ideas and some examples);
• Matrix and First-order Linear Systems: Eigenvalues and Eigenvectors; and
• Non-linear Autonomous Systems: Critical Point Analysis and Phase Plane Diagrams.

Prerequisites/Corequisites:
Single Variable Calculus is a prerequisite; Multivariable Calculus is recommended as a prerequisite or corequisite.

Texts: None required, but two good optional texts are:
(1) Elementary Differential Equations with Boundary Value Problems. 6th ed. by Edwards, C., and D. Penney. Upper Saddle River, NJ: Prentice Hall, 2008. ISBN: 9780136006138. [Note: The 5th Edition (ISBN: 9780131457744) or the 4th Edition will serve as well.]

(2) Differential Equations & Linear Algebra by Farlow, Hall, McDill, West. This text is published by Pearson and has ISBN #9780131860612.

Lecture Notes and Supplementary Materials     Course Calendar (with links to notes and supplements)

Essential Skills
Students should strive for personal mastery over the following skills. This list may be modified as the course proceeds.

• Model a simple system to obtain a first order ODE. Visualize solutions using direction fields and isoclines, and approximate them using Euler’s method.
• Solve a first order linear ODE by the method of integrating factors or variation of parameters.
• Calculate with complex numbers and exponentials.
• Solve a constant coefficient second order linear initial value problem with driving term exponential times polynomial. If the input signal is sinusoidal, compute amplitude gain and phase shift.
• Compute Fourier coefficients, and find periodic solutions of linear ODEs by means of Fourier series.
• Utilize Delta functions to model abrupt phenomena, compute the unit impulse response, and express the system response to a general signal by means of the convolution integral.
• Find the weight function or unit impulse response and solve constant coefficient linear initial value problems using the Laplace transform together with tables of standard values.
• Calculate eigenvalues, eigenvectors, and matrix exponentials, and use them to solve first order linear systems of ODEs. Relate first order systems with higher-order ODEs (reduction of order).
• Determine the qualitative behavior of an autonomous nonlinear two-dimensional system by means of an analysis of behavior near equilibria.

Homework:
Problem sets will be assigned each week and will be due the following week submitted online as a scanned PDF via Canvas. You are encouraged to discuss the homework with your fellow students, but you must write up the solutions by yourself without collaboration with others. (This is simply a matter of professional ethics.) Grading policies for the homework will be established after the first class meeting. Homework assignments and solutions will be posted on the course Calendar: http://math.rwinters.com/E21c/calendar.htm.

Solutions to the homework problems in PDF format will be made available via a password-protected web page linked from the Math E-21c course website. Selected problems may also be discussed in the problem sessions. Homework submitted after the posted deadline will be accepted only at the discretion of the Teaching Assistants.

Exams:
There will be two midterm exams and a two-hour final examination. All exams will be conducted online within Canvas using Proctorio. Details will be posted on the course website.

Grading:
The final grade will be based on the following scheme (subject to minor modification):
25% homework, 40% hour exams, 35% Final Exam

Note: The “Graduate” credit option is primarily for students enrolled in certain Extension School graduate programs such as the “Math for Teaching” program. All other students (including high school students) should register for the “Undergraduate” option or the Noncredit option (if you will not be submitting homework or taking exams). Students registered for the Graduate credit option may be asked to submit additional work beyond the basic homework assignments.

Topics and Assignments are posted in the Course Calendar.

Important Dates - Harvard University Extension School - Fall 2024

Official Academic Calendar

The Harvard Extension School is committed to providing an accessible academic community. The Accessibility Office offers a variety of accommodations and services to students with documented disabilities.

Please visit https://www.extension.harvard.edu/resources-policies/resources/disability-services-accessibility for more information.

You are responsible for understanding Harvard Extension School policies on academic integrity(https://www.extension.harvard.edu/resources-policies/student-conduct/academic-integrity) and how to use sources responsibly. Not knowing the rules, misunderstanding the rules, running out of time, submitting the wrong draft, or being overwhelmed with multiple demands are not acceptable excuses. There are no excuses for failure to uphold academic integrity. To support your learning about academic citation rules, please visit the Harvard Extension School Tips to Avoid Plagiarism (https://www.extension.harvard.edu/resources-policies/resources/tips-avoid-plagiarism), where you’ll find links to the Harvard Guide to Using Sources and two free online 15-minute tutorials to test your knowledge of academic citation policy. The tutorials are anonymous open-learning tools.

In particular: “To avoid any suggestions of improper behavior during an exam, students should not communicate with other students during the exam. Neither should they refer to any books, papers, or use electronic devices during the exam without the permission of the instructor or proctor.”

“Breaches of academic integrity are subject to review by the Administrative Board and may be grounds for disciplinary action, up to and including requirement to withdraw from the Extension School and suspension of registration privileges.”

Your lowest HW score (as a %) will be dropped when determining course grades.

Less Bronx-Whitestone Bridge Yielded More Stability During Hurricane Sandy (NY Times; Jan 4, 2013)
A success story of the sister bridge of the Tacoma Narrows Bridge

How do you re-engineer a suspension bridge to prevent collapse due to resonance? There’s a way... and there’s also a better way.

Whitestone Bridge with trusses (1946-2004)

Whitestone Bridge with trusses removed (2004-present)

References:

Subspaces, Span, Linear Independence, Basis of a Subspace; Images and Kernel of a Matrix

General Linear Spaces (Vector Spaces) and Solutions of ODEs

Laplace Transform Facts

Notes on Convolution

Matrix Methods for Solving Systems of 1st Order Linear Differential Equations

Sampler of Phase Plane diagrams for uncoupled, coupled, period, and stable sink linear systems and a nonlinear system

Supplement on Evolution Matrices

Supplement on Linear Coordinates, Alternate Bases, and Evolution Matrices

For your viewing pleasure: Arthur Mattuck explains numerical methods for solving ordinary differential equations (Euler’s Method)
Numerical Methods: Euler’s Method (MIT Open Courseware Video);  Example of Euler’s Method (MIT Open Courseware Video)

Arthur Mattuck, professor emeritus of mathematics, dies at 91
A specialist in algebraic geometry, the long-standing professor and former department head was influential across the Institute as an innovator in teaching first-year students.

Nov 2, 2021 – Arthur was my friend since 1978 and, once upon a time, a hiking partner on trips to the White Mountains. We would often run into each other at MIT concerts. Farewell, my friend. - RW

If ever you need to get in touch, you can contact me at either robert@math.rwinters.com or Robert@rwinters.com.

Nonlinear Systems and Linearization

Lorenz System, Lorenz Attractor (Wikipedia)

Mathematical Theory of Epidemics (Kermack, McKendrick, 1927) - 22 pages, somewhat technical, see pgs 713-714 in particular)

Here’s something:  http://micro.magnet.fsu.edu/primer/java/scienceopticsu/powersof10/index.html

Please send comments to Robert Winters.
URL: http://math.rwinters.com/E21c
Last modified: Tuesday, December 31, 2024 3:48 PM