Math E-21c - Ordinary Differential Equations - Fall 2024

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.

Course Meeting Times:
Lectures (Robert Winters): Wednesdays, 8:10pm-10:10pm via Zoom (subject to change)
Optional Recitations (by Teaching Assistants Renée Chipman, Kris Lokere, Jeremy Marcq)

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.

Course website:  http://math.rwinters.com/E21c and the related Canvas site (for accessing Lectures and recordings, submitting HW, and accessing HW and exam grades)

Prerequisites/Corequisites:
Single Variable Calculus is a prerequisite; Multivariable Calculus is recommended but not required; some familiarity with Linear Algebra is helpful but not required.

Texts: There is no specific required text for the course, but good references 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: Earlier editions and related editions of the Edwards & Penney text may also be good reference texts.]

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

We will provide our own Lecture Notes - which may be revised as the course proceeds. Additional supplements may also be provided.

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. Relate first order systems with higher-order ODEs.
• Determine the qualitative behavior of an autonomous nonlinear two-dimensional system by means of an analysis of behavior near critical points.

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% midterm 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 will be asked to submit additional work beyond the basic homework assignments.

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.

Academic Integrity: You are responsible for understanding Harvard Extension School policies on academic integrity and how to use sources responsibly. Violations of academic integrity are taken very seriously. Review important information on academic integrity and student responsibilities here: https://extension.harvard.edu/for-students/student-policies-conduct/academic-integrity; for more on academic citation rules, visit Using Sources Effectively and Responsibly (https://extension.harvard.edu/for-students/support-and-services/using-sources-effectively-and-responsibly) and review the Harvard Guide to Using Sources (https://usingsources.fas.harvard.edu).

Publishing or Distributing Course Materials: Students may not post, publish, sell, or otherwise publicly distribute course materials without the written permission of the course instructor. Such materials include, but are not limited to, the following: lecture notes, lecture slides, video, or audio recordings, assignments, problem sets, examinations, other students’ work, and answer keys. Students who sell, post, publish, or distribute course materials without written permission, whether for the purposes of soliciting answers or otherwise, may be subject to disciplinary action, up to and including requirement to withdraw. Further, students may not make video or audio recordings of class sessions for their own use without written permission of the instructor.

Proposed week-by-week syllabus (This may change somewhat as the course proceeds.)