Courses:

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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.

18.02 or 18.022 or 18.023 or 18.024 (corequisite), 18.01 or 18.014 (prerequisite).

The lecture period will be used to help you gain expertise in understanding, constructing, solving, and interpreting differential equations. You must come to lecture prepared to participate actively. At the first recitation you will be given a set of flashcards. Bring these with you to each lecture. (Extras will be available in lecture in case of need.) You will use them to announce your answer to questions posed occasionally in the lecture. In case of divided opinions a discussion will follow. As a further element of your active participation in this class, you will often be asked to spend a minute responding to a short feedback question at the end of the lecture. Despite the large size of this class, I will listen and respond to this feedback.

Edwards, C., and D. Penney. Elementary Differential Equations with Boundary Value Problems. 5th ed. Upper Saddle River, NJ: Prentice Hall, 2003. ISBN: 013145773X.

The Fourth Edition ( ISBN: 0130113018) will serve as well, and I will give reference numbers to both. The publisher has bundled Polking, Ordinary Differential Equations using MATLAB® with it at no extra cost. This is quite a good introduction to MATLAB®, but it will not be used in this course.

Students will also receive two sets of notes "18.03: Notes and Exercises" by Arthur Mattuck, and my "18.03 Supplementary Notes."

These small groups will meet twice a week to discuss and gain experience with the course material. Even more than the lectures, the recitations will involve your active participation. Come prepared. Your recitation leader may begin by asking for questions, so be ready if you have them. He may then hand out problems for you to work on in small groups. Ask questions early and often. Your recitation leader will also hold office hours, a resource you should not overlook.

Another resource of great value is the tutoring room. This is staffed by experienced undergraduates. Extra staff is added before hour exams. This is a good place to go to work on homework.

You should strive for personal mastery over the following skills. These are the skills that other courses at MIT will expect you to have when you finish 18.03. This list of skills is widely dissminated among the faculty teaching courses listing 18.03 as a prerequisite. You must become proficient at them to prepare yourself for those courses.

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).

The final grade will be based on a cumulative total of 900 points computed as follows:

Each problem set will be worth either 48 or 64 points, giving a total of 544 points. We will allow one missed problem set by replacing the problem set with the lowest score with the average score from the remaining problem sets. The total score will then be rescaled to 480, and half this score will be the contribution to the cumulative total.

Assignments will be due on Wednesdays or Fridays by 1:00. Each homework assignment 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 will be keyed closely to the lectures, and you 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. Your recitation leader should have the graded problems sets available for you at recitation on the Tuesday after they have been turned in. Solutions will be available on the afternoon of the day they are due, so late homeworks are not acceptable.

I encourage collaboration in this course, but I insist on honesty about it. If you do your homework in a group, be sure it works to your advantage rather than against you. Good grades for homework you have not thought through will translate to poor grades on exams. You must turn in your own write-ups of all problems, and, if you do collaborate, you write on your solution sheet the names of the students you worked with. Failure to do so constitutes plagiarism.

There are 3 one-hour exams held during the lecture hours.

There is 1 three-hour comprehensive final examination.

We will employ a series of specially written Java™ applets, or Mathlets. You will see them used in lecture occasionally, and each problem set will contain a problem based around one or another of them.