DIFFERENTIAL EQUATIONS FOR ENGINEERING AND PHYSICS
FINAL EXAM: Wednesday, December 20, 12:00–3:00 PM, Hill Center 116
- Here is the review problem set for the final
exam. As explained on the problem set, it is not complete,
covering primarily material we have covered since the second exam. You
should also consult the review problem set for exam
1 and the review problem set for exam 2.
- The problem discussion session for the final will be Thursday,
December 14, from 2:30 to 4:30 in Hill 116.
- I will hold office hours in Hill 520 Monday, 12/18,
10:00–11:30 AM and Tuesday, 12/19, 3:30–5:00 PM.
- I have I have now posted all the course grades available—ten
quizzes, four Maple labs, and two midterms—on the SAS
Gradebook. You are welcome to check these grades for errors.
The login procedure is straightforward. If you have trouble with the site,
or believe that one or more of your grades has been recorded improperly,
get in touch with me. I will post the Maple Lab 5 grade once it is
available; I will also post your final exam and course grades there.
- Here are the phase plane plots for the predator-prey model which
were shown in class today. These plots use the equations (1) of Section
9.5 of Boyce and DiPrima, with the parameter values a = 1.4 , c = 2.0 ,
α = 0.8 , γ = 1.2 .
- The first quadrant of the
phase plane, showing the direction field, the
nullclines, and some solution curves. Solution curves were obtained by
graphing the level curves of the conserved quantity H(x,y), not by solving
the system of differential equations.
- A pair of solution curves, showing the prey
population (blue) and predator population (red) as functions of t . The
initial populations were both taken to be 1. Notice that the solutions are
periodic and that the predator oscillations lag behind the oscillations of
- A second pair of solution curves, drawn with
the same conventions as above. The initial populations were both taken to
be 0.5, leading to larger oscillations.
- One does in fact observe cyclic behavior of populations in nature, but
it is not clear to what extent a simple model like the predator-prey
equations we have studied can help explain them. If you would like to read
more about this, here are a (very) few references. I displayed pages from
the first and third of these in the lecture.
- An article by C. Elton and M. Nicholson
discussing in some detail the actual records for lynx pelts of the Hudson
Bay Trading Company. There are nice graphs showing the cyclic oscillations
in the lynx population (as indicated by trapping records).
- A survey article by P. J. Wangersky
discussing the general applicability of populations models of the Lotka-Volterra type.
- An article by M. E. Gilpin entitled
"Do Hares Eat Lynx?" Besides its amusing title, this article seems to have
intelligent things to say about the general problem of modeling population
- A set of
lecture notes from a course given by Dr. Bob Schooley at the University
of Illinois. This is a power point presentation. Here is the course web page with more
- Maple lab 5 is now posted below; the lab is due Monday 12/11.
- Here are the phase plane plots for the competing species models which
were shown in class today. Recall that in these plots red lines
denote vertical nullclines, blue lines denote horizontal nullclines,
and black lines denote trajectories.
- As announced in class, we changed the syllabus, removing Section 7.8
and substituting Section 7.7. That means that you should ignore the
homework problems assigned for Section 7,8. For Section 7,7, do the
following problems (to be discussed in recitation Tuesday, 11/28):
1, 3, 6, 11, 12, 14. Remember that what the book calls "the fundamental matrix
Phi(t) satisfying Phi(0)=I" is what we called the "matrix exponential exp(At)".
- The writeup for Maple lab 4 had a couple of unwanted references, in
sections 2 and 3, to the supplementary worksheet; these have now been
- Maple lab 4 is now available below; the lab is due in lecture Monday
- The second exam is Wednesday, November 15. The exam will cover
through Section 7.5, including material on linear algebra which is in the
notes on linear algebra but not in the text
(this is the schedule originally announced). The emphasis will be on
material since the last exam - that is, from Section 3.5 onward - but some
material from earlier in Chapter 3 may creep in. Chapters 1 and 2 will not
be on the exam.
- Here is a set of review problems for Exam
- We will have a problem/review session Sunday, October 12, at
7:30 PM in Hill 116. Bring questions: if you don't ask me anything the
session will be over quickly.
- Here are the pictures of the phase plane
for a saddle point and for a stable node that I showed in class; there is
also a picture of the unstable spiral point that we will discuss Monday.
- Here are some further notes. These
are just a summary of the important general features of solutions of linear
systems of differential equations. Everything in the notes is already in
Section 7.4 of Boyce and DiPrima, but perhaps it will be helpful to have a
- Here are some notes on linear algebra,
covering material that I discussed 10/25, 10/30, and 11/01. These
should be regarded as a supplement to Section 7.3 of Boyce and
DiPrima; you should read that section as well as the notes. The notes also
contain some exercises.
- Revised homework assignment for Tuesday, November 7:
Section 7.2, problem 11;
Section 7.3, problems 1,2,3, 6, 10, 12, 13, 15, 18, 19, 21, 22, 24, 28;
Notes, exercises 1, 2, 3, 4.
- Today and next week I will be talking about linear algebra,
including some material that is not in the text. I will provide
some notes here soon, but please be aware that you are responsible for
material covered in lecture, even if it is not in the text.
- The homework assigned for Section 7.2 (to be discussed Tuesday,
October 31) has been modified. The assignment is now problems 2, 3, 4, 8,
22, and 26.
- Maple lab 3 is now available below; it is due Monday,
- I postponed the due date for Maple lab 2 until Wednesday, October 25.
The lab is available below. The due date for Lab 3 will be
Monday, November 6.
- The next Maple lab, Lab 2, is due Monday, October 23. It is available
- We are now back on the schedule originally announced for the class, so the
quiz this week will cover all material since the last quiz: Sections 3.7,
3.8, and 3.9.
- The first exam is Wednesday, October 10. The exam will cover
through Section 3.4. Note that Section 3.5 will not be on the
exam; this is a change from the schedule originally announced.
- Here is a set of review problems for Exam
1. Please note these changes:
- Omit problem 2(b).
- Correct the answer for problem 1(d) to (-1.60337,1.60337).
- We will have a problem/review session Sunday, October 8, at 8:00 PM
in Hill 116. Bring questions: if you don't ask me anything the session
will be over quickly.
- Here are the direction field and solution
plots for the logistic and exponential growth equations, as shown in
class on 9/20.
- It was pointed out to me that in most of the assigned problems in the
first part of Section 2.6 the differential equation is not exact; this
means that the problems don't furnish much practice in finding the
solution. Try some other problems: Section 2.6 3, 5, 9.
- We will omit the backward Euler method
discussed in Section 8.1.
- Remember that Maple lab 0 will be collected (and checked off,
but not graded) on Wednesday, September 20.
Note change: Aek will hold his office hours Monday 11:00-12:00 and
Thursday 3:00-4:00 in Hill Center 512.
Information and help
- Some general
instructions to be used in Math 251 in Fall 2006. It is in Math 251
that most Rutgers students taking mathematics first encounter Maple; the
instructions here contain a lot of general information about the program.
An introduction to Maple
prepared by Professor S. Greenfield. This is a good starting place if you
have not used Maple before.
- Maple help pages
prepared by Rutgers TA L. Pudwell. They were written with Math 251
students in mind, but should be helpful for all.
- An introduction to Maple features
relevant to differential equations, with a worksheet illustrating these features. These
documents are almost identical to those prepared for Spring 2006 by
Professor Bumby. A separate document showing new features of Maple 10,
which was available in Spring 2006, has been absorbed into the Math 251
Maple instructions referred to above.
Maple labs for 244:06-08
Your Maple labs should have a title, your name, and your
section number at the top. They must be stapled. I will
not accept labs submitted by email.
- Lab 5, with the corresponding
seed file. Lab 5 is due in lecture
Monday, December 11. Much of this lab is new; it may be somewhat
longer and more difficult than the earlier labs. I would welcome any
comments that you have.
- Lab 4, with the corresponding
seed file as well as a
supplementary file. Lab 4 is due in lecture
Monday, November 27.
- Lab 3, with the corresponding seed file. Lab 3 is due in lecture
Monday, November 6.
- Lab 2, with the corresponding seed file. Lab 2 is due in lecture
Monday, October 23.
- Lab 1, with the corresponding seed file. Lab 1 is due in lecture
Wednesday, October 4.
- Lab 0, with the corresponding seed file.