Math 252-02 Spring 2000
Special information for Prof. Bumby's section will appear on this
page. Common features of all sections appear on the course page. Students are encouraged to explore the
Information and Resources links on the course history page
Schedule of quizzes and exams
- Friday, January 28: Quiz on 1.1 (modeling) and 1.2 (separable
- Wednesday, February 09: Quiz on Separable Equations (again) and
Slope Fields. The first project will be distributed at this time.
When a course using this textbook was first developed, the projects
were intended to give experience in modeling. The suggested first
project in the course history page has replaced the original
project with one similar to a Maple project used in our 244
course. This section will use a version of the original modeling
- Friday, February 18: Quiz on 1.6 (phase line), 1.7 (bifurcation),
and 1.8 (linear equations).
- Wednesday, February 23: Class will meet in the Computer lab
ARC-118 instead of regular classroom to allow discussion of use of
Maple to illustrate all aspects of the course. Project 1 due
today if you do it without Maple. An additional week will be
allowed if you use Maple since there was no previous
discussion of that system. The Maple worksheets shown during
that class can be found in the ~bumby directory on eden. You should
be able to copy all relevant files.
- Friday, February 25: Exam 1 -- chapters 1 and 2.
- Friday, March 10: so late on the eve of Spring Break, new topics
are not appropriate, so class will be devoted to questions about the
notes on matrix exponentials N4 and its
application to chapter 3 of the text, along with any other old
- Wednesday, March 22: project 2 will be distributed.
- Friday, March 24: quiz on matrix exponentials applied to solution
of linear systems, with emphasis on case of distinct real
- Wednesday, April 5: another quiz on matrix exponentials applied to solution
of linear systems, with emphasis on other cases.
- Wednesday, April 12: Exam on chapters 3 and 4. Very little of
chapter 4 will have been covered in lecture in time to be a
significant part of this exam, but the beginning of the chapter helps
to focus the results of chapter 3 on an important application.
- Wednesday, May 10, Noon to 3PM: Final exam. About half of the
exam should resemble questions from the hour exams. The additional
topics, which seemed few when I was planning the exam, expanded to
take a significant amount of space. These topics are: (1) identifying
equilibrium points of autonomous nonlinear systems and selectively
representing information to find qualitative aspects of solutions.
This topic was introduced at the time of the first exam and played a
role in Project 2 and the brief visit to chapter 5 of the text late in
the course. Such questions always suggest more than needed to give
complete answers to what is requested, but the extra information can
be used to detect misconceptions in your initial analysis. Don't get
carried away searching for properties of the equation, but aim for
accuracy in all conclusions. (2) The treatment of harmonic oscillators
in chapter 4 was introduced at the time of the second hour exam.
There are two parts to the solution of these equations. The
homogeneous part illustrates the results of chapter 3, but has a
slightly different emphasis; the inhomogeneous part is another example
of the power of a good guess that was a regular theme in the text and
supplementary notes. The importance of the problem in applications
has led to formulas for solving the equation quickly. If you can use
these formulas to get correct answers, they will be accepted,
but the exam problem should be solvable by applying general princples.
(3) Properties of Discrete Dynamical Systems from chapter 8 discussed
during the last week of the course. The emphasis will be on
experimenting with iteration, and a programmable calculator (with the
ability to program it) will be helpful. The characterization of
attracting, repelling and neutral fixed points (and cycles) derived
(though not stated as a theorem) on page 616 is the basis of our
understanding of iteration, so it can be expected to be a main feature
of the exam questions in this area.
- N1 revised form of modeling
- Transcript of a Maple session
generating slope fields.
- Euler's method notes of lecture of
- Bifurcation notes in PDF format,
that should be easier to print, but otherwise close to the comments on bifurcations in HTML
format from the Additional
- Project 1 as distributed in class on
- Example discussed in class of
Friday, February 11.
- Linearization notes prepared by
another instructor to illustrate examples from sections 2.1 and 2.2 of
- N3 (slightly revised, in pdf
format, but the correction of an obvious error in the first example
must wait for a while pending access to the source file).
- N4 (slightly revised, in pdf
- Maple worksheets based on examples appearing in the exercises
for section 2.4 (announced in class on March 8). You should
"shift-click" to download these files to your directory, and then
view them in xmaple. Since they will now be your
worksheets, you can modify them and try other experiments
suggested by what they include. The worksheets deal with exercise 5 and exercise
- Project 2 as distributed in class on
- Complex eigenvaluesanalyzed using
matrix exponentials. Notes prepared by another lecturer.
- Skew symmetric matrices and the
systems that give rise to them. An extension of the previous
- An easily remembered formula for
solutions of 2 by 2 systems with a pair of complex conjugate
eigenvalues. Details of the derivation of the formula given in the
class of March 29.
- Variation of Parameters used to
solve inhomogeneous equations. Notes prepared by another lecturer.
Notes and Comments
The slope fields shown in class on Wednesday, January 26 were
obtained in the transcript mentioned in the Supplementary
Material section. The cause of the similarity of pictures for
exercise 3 and exercise 6 was not noticed until the slides were shown
in class, so both appear in the transcript. The instructions used is
slightly different from the one described in the 1-dim direction fields help
page, but both work. Although Maple is not a formal part of
this course, pointers will be given on using it to illustrate material
from lectures and textbook.
Projects are expected to take two weeks. Project
1, distributed on February 09, was based on a similar project
used in Fall 1998, but there are minor changes in all parts. The
project from Fall 1999 in the course archives is based on a
Maple lab project from Math 244 and has a very different
format. Since the projects and instructions from Math 244 may contain
hints that will be useful here, here is an easy link to those
files. The tutorial portions of those labs will give examples of
common tools for working with differential equations, even if you skip
the embedded exercises. Project 2, distributed on
March 22 is new this term. It uses the Van der Pol equation to as a
vehicle for reviewing many topics from previous courses. The normal
time limit of two weeks may need to be extended. One extra week
should suffice: the open-ended parts of project are only intended to
provide a goal to the work, allowing some discussion of the more
routine exercises. You may not be able to do everything suggested in
the project in the available time.
Comments on this page should be sent to:
Last updated: April 20, 2000