Spring 2020
Differential Equations
Math 18.03, Concourse - MIT

Lectures by: Robert Winters
Office: 16-137
Phone: x3-2050 (but e-mail is better)
Alternate email: robert@math.rwinters.com

Recitations by: Robert Winters

Calendar of topics and
homework assignments

Notes and Supplements

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Concourse Stellar site


Concourse 18.03 Stellar site

Though there is no required text for the course, two texts that have been used for the course in the past may serve as useful references. They are:

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.
[Click on the image below for online prices.]
Differential Equations & Linear Algebra

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

Elementary Differential Equations with Boundary Value Problems, 6th Edition by Edwards & Penney (ISBN 0130339679 for 2008 hardcover edition) or an earlier edition. This text was used for the Spring 2013 and earlier semesters.
Elementary Differential Equations with Boundary Value Problems, 6th Edition by Edwards & Penney

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

Earlier editions of the Edwards & Penney text would also be good reference texts.


Lecture #3 Notes and Lecture #4 Notes (we'll be intermixing from both this week)
Lecture #5 Notes (we may also draw from this a bit later in the week)

Problem Set #2 (due Wed, Feb 19)
References: EP 1.5 (Linear 1st Order Equations); IR1–3 (Input-Response Models); SN 2 (Modeling by 1st Order Linear ODEs); SN 3 (Solutions of 1st Order Linear ODEs)

Lecture #2 Notes (revised Feb 5)

Lecture #1 Notes (revised Feb 2)

Problem Set #1 (due Tues, Feb 11)
References: EP 1.1 (Differential equations and mathematical models); EP 1.2 (Integrals as general and particular solutions); EP 1.3 (Slope fields and solution curves); EP 1.4 (Separable equations and applications); Notes D (Definite Integral Solutions); Notes G.1 (Graphical & Numerical Methods); SN 1 (Notation & Language); EP 1.7 (Population models); EP 7.1 (Equilibrium solutions and stability); Isoclines applet; SN 2 (Modeling by 1st Order Linear ODEs)

We will again be offering this course during the Spring 2020 semester. We'll be following the lead of the mainstream 18.03 course and not have a specific required text. However, you may still want to use a reference text. See the left column and links for recommended texts. All problem sets will be made available as printable PDFs. Much of the course material will be derived from the 18.03 Supplementary Notes by Prof. Haynes Miller (used in recent years in the mainstream version of this course), the 18.03 Notes and Exercises by Prof. Arthur Mattuck, and Lecture Notes written specifically for our course.

Syllabus for Concourse Math 18.03 (Spring 2020)

Printable syllabus (PDF, Spring 2020)

Lectures are on Mondays and Wednesdays from 12:00pm to 1:30pm in 16-160.

There are two recitations (in order to accommodate conflicting schedules):
Tues, Thurs from 11:00am to 12:00pm in 16-160
Tues, Thurs from 2:00pm to 3:00pm in 16-160
[I will also be available for anyone who missed recitation on Tues, Thurs from 12:00pm to 1:00pm.]

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
Whitestone Bridge with trusses (1946-2004)

Whitestone Bridge without trusses
Whitestone Bridge with trusses removed (2004-present)

Concourse Mathematics Tutoring Schedule
Tutor/Grader Course Day, Time
Sophie Cohen 18.03 --
Kenny Cox 18.03 --
Jasmine Yang 18.03 --


ODE Manipulatives ("Mathlets")
This course employs a series of specially written Java™ applets, or Mathlets, developed by the Mathematics Department. Each problem set typically contains a problem based around one or another of them. [Full catalog of Mathlets from Math Dept.]

18.03 Supplementary Notes by Prof. Haynes Miller (used in the mainstream version of this course)

18.03: Notes and Exercises by Prof. Arthur Mattuck

Additional Notes

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

General Linear Spaces (Vector Spaces) and Solutions of ODEs (RW)

Laplace Transform Facts

Notes on Convolution (RW)

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

Phase portraits for the linear ODE examples

Supplement on Evolution Matrices

Supplement on Linear Coordinates, Alternate Bases, and Evolution Matrices

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

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

18.01 (Single Variable Calculus) is a prerequisite; 18.02 (Multivariable Calculus) is a corequisite, meaning students may take 18.02 and 18.03 simultaneously.

Texts: None required, but two good optional textx are:
(1) Differential Equations & Linear Algebra by Farlow, Hall, McDill, West. This text is published by Pearson and has ISBN #9780131860612.

(2) 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.]

Students will also need two sets of notes "18.03: Notes and Exercises" by Arthur Mattuck, and "18.03 Supplementary Notes" by Haynes Miller (both available online at no cost). We will primarily use these notes and our own Lecture Notes which may be revised as the course proceeds. Additional supplements may also be provided.

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

  • Solution of First-order ODE's by Analytical, Graphical and Numerical Methods;
  • Linear ODE's, Especially Second Order with Constant Coefficients;
  • 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;
  • Matrix and First-order Linear Systems: Eigenvalues and Eigenvectors; and
  • Non-linear Autonomous Systems: Critical Point Analysis and Phase Plane Diagrams.

The Concourse version of the 18.03 course will closely parallel the mainstream 18.03 course. As has been the case for the last few semesters, there will be additional emphasis on Linear Algebra throughout the course, and some topics listed above may be less emphasized than in previous years.

The lecture period is used to help students gain expertise in understanding, constructing, solving, and interpreting differential equations. Students must come to lecture prepared to participate actively. Students may sometimes be asked to spend a minute responding to a short feedback question at the end of the lecture.

These meet twice a week to discuss and gain experience with the course material. Even more than the lectures, the recitations involve active participation. The recitation leader may begin by asking for questions or hand out problems to work on in small groups. Students are encouraged to ask questions early and often.

Office Hours
Regular office hours at times to be determined. You are encouraged to drop by for any matters that cannot adequately be addressed in class.

Tutors/graders are available within Concourse. Another resource of great value to students is the Mathematics Department tutoring room. This is staffed by experienced undergraduates. This is a good place to go to work on homework (as is the Concourse Lounge).

You may find the 18.03 lecture videos of Arthur Mattuck helpful. They are available on the Open Courseware site and were recorded in Spring 2003.

The Ten Essential Skills
Students should strive for personal mastery over the following skills. These are the skills that are used in other courses at MIT. This list of skills is widely disseminated among the faculty teaching courses listing 18.03 as a prerequisite. At the moment, 140 courses at MIT list 18.03 as a prerequisite or a corequisite.

  1. Model a simple system to obtain a first order ODE. Visualize solutions using direction fields and isoclines, and approximate them using Euler's method.
  2. Solve a first order linear ODE by the method of integrating factors or variation of parameter.
  3. Calculate with complex numbers and exponentials.
  4. 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.
  5. Compute Fourier coefficients, and find periodic solutions of linear ODEs by means of Fourier series.
  6. 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.
  7. 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. Relate the pole diagram of the transfer function to damping characteristics and the frequency response curve.
  8. Calculate eigenvalues, eigenvectors, and matrix exponentials, and use them to solve first order linear systems. Relate first order systems with higher-order ODEs.
  9. Recreate the phase portrait of a two-dimensional linear autonomous system from trace and determinant.
  10. Determine the qualitative behavior of an autonomous nonlinear two-dimensional system by means of an analysis of behavior near critical points.

The Ten Essential Skills is also available as a (PDF).

Homework assignments typically will have two parts: a first part drawn from the book or notes, and a second part consisting of problems which will be handed out. Both parts are keyed closely to the lectures. Students should form the habit of doing the relevant problems between successive lectures and not try to do the whole set the night before they are due.

There are 3 one-hour exams held during lecture session and a three-hour comprehensive final examination.

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

ODE Manipulatives ("Mathlets"):
This course employs a series of specially written Java™ applets, or Mathlets, developed by the Mathematics Department. They may be used in lecture occasionally, and each problem set typically contains a problem based around one or another of them.

Topics and Assignments are posted in the Course Calendar.

Quantum Sandwich

If ever the MIT mail servers are not accessible from the outside world and you need to get in touch, you can also contact me at either robert@math.rwinters.com or Robert@rwinters.com.

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

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Please send comments to Robert Winters.
URL: http://math.rwinters.com/1803
Last modified: Tuesday, February 11, 2020 0:01 AM