Spring 2017
Differential Equations
Math 18.03, Concourse - MIT

Lectures by: Robert Winters
rwinters@mit.edu
Office: 16-137
Phone: x3-2050 (but e-mail is better)

Recitations by: Robert Winters
(with occasional assistance of other wizards)

Calendar of topics and
homework assignments

Notes and Supplements

Solutions
(username/password required)


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.

Announcements:

The Final Exam took place on Thursday, May 25 from 1:30pm to 4:30pm in Room 16-160. Course grades will be submitted soon.

Approximate letter grades for Exam #1
Total points on exam was 50. Median score was 50 (100%).
Mean score was 46.0 (92%). Standard deviation was 6.0.
score grade   score grade
47+ A 32+ C+
45+ A– 29+ C
42+ B+ 27+ C–
38+ B 25+ D
35+ B– 0-24 F
    
Approximate letter grades for Exam #2
Total points on exam was 60. Median score was 55 (91.7%).
Mean score was 48.8 (81.4%). Standard deviation was 15.0.
score grade   score grade
56+ A 37+ C+
52+ A– 34+ C
48+ B+ 31+ C–
44+ B 28+ D
41+ B– 0-27 F
Approximate letter grades for Exam #3
Total points on exam was 50. Median score was 44 (88%).
Mean score was 42.5 (84.9%). Standard deviation was 8.7.
score grade   score grade
46+ A 31+ C+
43+ A– 28+ C
40+ B+ 25+ C–
37+ B 23+ D
34+ B– 0-22 F
 
Approximate letter grades for Final Exam
Total points on exam was 110. Median score was 88.5 (80.5%).
Mean score was 86.5 (78.6%). Standard deviation was 18.6.
score grade   score grade
100+ A 67+ C+
92+ A– 62+ C
84+ B+ 57+ C–
78+ B 52+ D
72+ B– 0-51 F

Exam #3 took place during class on Wed, May 3.

Exam #2 took place on Thurs, April 6.

Exam #1 took place on Tuesday, March 7.


Problem Set #10 (due Fri, May 12)
References: Notes LS.3 (Complex and repeated eigenvalues); EP 7.2 (Stability and the phase plane); Notes GS.1–5 (Graphing ODE systems); Supplement on Evolution Matrices; Supplement on Linear Coordinates, Alternate Bases, and Evolution Matrices ; EP 5.7 (Matrix exponentials and linear systems); Notes LS.6 (Solution matrices); Matrix Methods for Solving Systems of 1st Order Linear Differential Equations; Phase portraits for the linear ODE examples


Problem Set #9 (due Tues, May 9)
References: EP 5.1 (First-Order systems and applications); EP 5.2 (The method of elimination); EP 5.3 (Matrices and linear systems); Notes LS.1 (Linear systems: Review of linear algebra); Notes LS.2.2 (Homogeneous linear systems w/constant coefficients); EP 5.4 (The eigenvalue methods for homogeneous systems); SN 30 (First order systems and second order equations); SN 31 (Phase portraits in two dimensions); Supplement on Evolution Matrices; Supplement on Linear Coordinates, Alternate Bases, and Evolution Matrices; Matrix Methods for Solving Systems of 1st Order Linear Differential Equations; Phase portraits for the linear ODE examples


Problem Set #8 (due Fri, Apr 28)
References: SN 25 (Laplace Transform technique: coverup); SN 26 (The Laplace Transform and generalized functions); Notes H (Heaviside coverup method); EP 4.3 (Translation and partial fractions); Laplace Transform Facts; Notes on Convolution (RW); FH 8.4 (The Convolution Integral and the Transfer Function); Notes CG (Convolutions and Green's formula); SN 24 (Convolution); SN 27 (The pole diagram and the Laplace Transform); SN 28 (Amplitude response and the pole diagram); SN 29 (The Laplace Transform and more general systems)


Problem Set #7 (due Thurs, Apr 20)
References: SN 21 (Steps, impulses and generalized functions); SN 22 (Generalized functions and differential equations); SN 23 (Impulse and step responses); Notes IR (Input response models); Laplace Transform Facts; EP 4.1 (Laplace transforms and inverse transforms); EP 4.4 (Derivatives, integrals, and products of transforms); EP 4.2 (Transformation on initial value problems); FH 8.1 (The Laplace Transform and its inverse); FH 8.2 (Solving DEs and IVPs with Laplace Transforms); FH 8.3 (The Step Function and Delta Function); Problem Set #7 Hints and Advice


Problem Set #6 (due Thurs, Apr 13)
References: EP 8.1 (Periodic functions and trigonometric series); SN 20 (More on Fourier series); EP 8.2 (General Fourier series and convergence); EP 8.3 (Fourier sine and cosine series); EP 8.4 (Applications of Fourier series); and Lecture Notes.


Problem Set #5 (due Thurs, Mar 23)
References: EP 2.6 (Forced oscillations and resonance); SN 12 (Resonance); Notes O (Linear differential operators); SN 11 (Undetermined coefficients); SN 13 (Time invariance); SN 14 (The exponential shift law); SN 15 (Natural frequency and damping ratio); Review class notes on Exponential Shift Formula and Variation of Parameters.


Problem Set #4 (due Thurs, Mar 16)
References: EP 2.1 (2nd Order Linear Equations); EP 2.2 (General solutions of linear equations); SN 19 (The Wronskian); EP 2.3 (Homogeneous equations w/constant coefficients); EP 2.4 (Mechanical vibrations); SN 7 (Beats); SN 8 (RLC circuits); SN 9 (Normalization of solutions); SN 10 (Operators and the exponential response formula); EP 2.5 (Nonhomogeneous equations and undetermined coefficients); 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)


Problem Set #3 (due Mon, Mar 6)
References: EP 1.7 (Population models); EP 7.1 (Equilibrium solutions and stability); Notes IR.6 (Input Response Models); SN 4 (Sinusoidal Solutions); SN 5 (Algebra of Complex Numbers); C.1-C.4 (Complex Numbers); SN 6 (The Complex Exponential)


Problem Set #2 (due Thurs, Feb 23) - See Course Calendar for more details and references.
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); and posted Lecture Notes.


Problem Set #1 (due Thurs, Feb 16) - See Course Calendar for more details and references.
The deadline was extended due to the snow cancellations, but if you have not yet submitted this p-set, you should do so NOW.


Lectures are on Mondays and Wednesdays from 1:30pm to 3:00pm in 16-160.

There are two recitations (in order to accommodate conflicting schedules):
Tues, Thurs from 11:00am to noon in 16-160
Tues, Thurs from 2:00pm to 3:00pm in 16-160

If there is sufficient demand, we may have an optional weekly informal session covering some more theoretical aspects of the course - possibly on Wednesday afternoons.


We will again be offering this course during the Spring 2017 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 2017)

Printable syllabus (PDF, Spring 2017)


Concourse Mathematics Tutoring Schedule
Tutor/Grader Course Day, Time
Kerrie Greene 18.03 --
Tinna-Solveig Kosoko-Thoroddsen 18.03 --
Ayomide Fatunde 18.03 --

Course Meeting Times:
Lectures (Robert Winters): Mondays and Wednesdays, 1:30pm to 3:00pm
Recitations (Robert Winters): Tues, Thurs from 11:00am to noon or 2:00pm to 3:00pm
An optional weekly session covering some more theoretical aspects of the course will also be scheduled.


References:

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)


Prerequisites/Corequisites:
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.

Description:
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.

Lectures
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.

Recitations
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.

Tutoring
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).

Videos
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:
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.

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

Grading:
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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URL: http://math.rwinters.com/1803
Last modified: Monday, May 29, 2017 1:10 AM