MIT Concourse - 18.03 Syllabus - Spring 2019

**Course Meeting Times**:

__Lectures__ (Robert Winters): Mondays and Wednesdays, 1:30pm-3:00pm

__Recitations__ (Robert Winters): Tuesdays and Thursdays __either__ 11:00am to noon or 2:00pm-3:00pm

**Course website**: http://math.rwinters.com/1803 and the related Stellar site (for HW and exam grades)

**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 texts are:

(1)

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

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.]Elementary Differential Equations with Boundary Value Problems. 6th ed.

We will also make use of "18.03: Notes and Exercises" by Arthur Mattuck, and "18.03 Supplementary Notes" by Haynes Miller (both available online at no cost) as well as lecture notes prepared specifically for our course. If applicable, we may also reference additional materials from the mainstream course.

**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__.

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 (less than in past years, but we'll still cover this);
- Matrix and First-order Linear Systems: Eigenvalues and Eigenvectors; and
- Non-linear Autonomous Systems: Critical Point Analysis and Phase Plane Diagrams.

__Lectures__

The lecture period is used to help students gain expertise in understanding, constructing, solving, and interpreting differential equations. Students should come to lecture prepared to participate actively.

__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. 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. Other good videos are available via OCW.

**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.

- 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 parameter.
- 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. Relate the pole diagram of the transfer function to damping characteristics and the frequency response curve.
- Calculate eigenvalues, eigenvectors, and matrix exponentials, and use them to solve first order linear systems. Relate first order systems with higher-order ODEs.
- Recreate the phase portrait of a two-dimensional linear autonomous system from trace and determinant.
- 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 consist of a combination of routine skill-based problems drawn from a textbook or notes, and other problems that may be more searching and interpretive. Both kinds of problems will be tied to topics presented in 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 will be 3 one-hour exams held during either a lecture class or a recitation. There will also be a three-hour comprehensive final examination.

**Grading**:

The final grade will be based on the following scheme (subject to minor modification):

25% homework, 40% hour exams, 35% 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.