Math E-21c - Ordinary Differential Equations - Fall 2021
This course is a study of Ordinary Differential Equations (ODE's), including modeling physical systems. Topics will likely include:
Course Meeting Times:
Lectures (Robert Winters): Wednesdays, 5:50pm-7:50pm via Zoom
Optional Recitations to be determined
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 also 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.
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.
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.
Proposed week-by-week syllabus (This may change somewhat as the course proceeds.)
Date |
Topics |
Sept 1 |
Basic notions, modeling with ordinary differential equations, separable equations, natural growth vs. logistic growth models, stability; 1st order linear ODEs, linear time-invariant ODEs, solution of 1st order linear ODEs using integrating factor. |
Sept 8 |
Newton's Law of Cooling; homogeneous vs. inhomogeneous solutions, method of undetermined coefficients; Variation of Parameters; higher order linear ODEs; signal-response (or input-response) perspective. |
Sept 15 |
Linear system response to exponential and sinusoidal input; gain, phase lag. Complex-valued equation associated to sinusoidal input. The algebra of complex numbers; the complex exponential; complex numbers, roots of unity. Applications to trigonometry, integration, and solving ODEs (complex replacement). Complex-valued equation associated to sinusoidal input; gain, phase lag. |
Sept 22 |
Basics of Linear Algebra: Subspaces, span, image and kernel, linear independence, basis, dimension, coordinates relative to a basis, function spaces and linear operators; 2nd order linear constant coefficient ODEs, characteristic polynomial, modes, independence of solutions, and superposition of solutions; Wronskian determinant; sinusoidal and exponential response; harmonic oscillator; complex characteristic roots. |
Sept 29 |
Linear operators with constant coefficients (time invariant), exponential solutions, characteristic polynomial, distinct real roots, complex roots, repeated roots; Hooke's Law; complex replacement; Exponential Response Formula (ERF) and the Resonance Response Formula (RRF) for exponential and sinusoidal input signals. Gain and phase lag; resonance and forced harmonic motion; RLC circuits. |
Oct 6 |
Exponential Response Formula (ERF), Resonance Response Formula (RRF); Resonance, Frequency response, LTI systems, superposition, RLC circuits [Resonance; Frequency response; RLC circuits; Time invariance]; Exponential Shift Rule; Variation of Parameters for higher order systems; Discontinuous inputs. |
Oct 13 |
Fourier series: orthogonality, inner products, orthogonal projection, Pythagorean Theorem; Applications to ODEs - harmonic response, resonance; Fourier series for periodic inputs; Fourier's Theorem and Fourier coefficients; square wave function; Sawtooth function; Differentiating and integrating Fourier series; Tips & Tricks: trig id, linear combination, shift |
Oct 20 |
Generalized functions, generalized derivative, step and delta functions; Impulse and step responses; Laplace transform: basic properties, rules and sample calculations; t-domain vs s-domain; idea of how to solve ODEs by translating differential equations into algebraic equations; Step and delta functions. Impulse and step responses, generalized derivative; unit impulse response; time invariance; |
Oct 27 |
Solution with initial conditions as w*q. Inverse transform; non-rest initial conditions for first order equations; worked examples of Laplace Transform and convolution |
Nov 3 |
Linear algebra: linear independence, span, basis, coordinates; matrix of a linear transformation relative to a basis; Introduction to vector fields and systems of 1st order ODEs; reduction of order - nth order equations and systems of 1st order equations; matrix representation. First order linear systems of ODEs in matrix form, solution of uncoupled (diagonal) systems and evolution matrices; uncoupling a system (diagonalization) in case of real eigenvalues, evolution matrices; Sampler of Phase Plane diagrams for uncoupled, coupled, period, and stable sink linear systems and a nonlinear system. |
Nov 10 |
Complex eigenvalues; Qualitative behavior of linear systems; phase plane; Solving System of 1st Order Linear Differential Equations, continued; repeated eigenvalues; decomposition of 1st order linear system into mode (block matrices); simple nonlinear system with shifted equilibrium. Matrix Methods for Solving Systems of 1st Order Linear Differential Equations; Phase portraits for the linear ODE examples. |
Nov 17 |
Qualitative behavior of linear systems; phase plane [Eigenvalues vs coefficients; Complex eigenvalues; Repeated eigenvalues; Defective, complete; Trace-determinant plane; Stability]; simple nonlinear systems. |
Dec 1 |
Nonlinear systems; linearization near equilibria, Jacobian matrices; the nonlinear pendulum; autonomous systems, predator-prey systems. |
Dec 8 |
Nonlinear systems; chaotic dynamical systems; Lorenz System, Lorenz Attractor; Mathematical Theory of Epidemics |
Dec 15 |
Final Exam |