MATH 44:
Differential Equations
Mathematics Department, Swarthmore
College, Spring 2011
Professor: Victor Donnay http://www.brynmawr.edu/math/people/donnay/ 
Lecture:
MWF 9:30 – 10:20 Possibility
this will change to
MW 9 – 10:20 
Office: SC #143 

Phone: 610 957 6292., Email: vdonnay1 
Office
Hours: tba 
Prerequisites: You should have taken Calculus of Several Variables/
Multivariable calculus and Linear Algebra. Please speak to me if you have not
had one or the other of these courses.
Text: Differential Equations by Blanchard, Devaney, and Hall, 3^{nd}
edition, published by Brooks/Cole.
Course Web Site: accessible
from Prof. Donnay's
homepage
All
materials for the course will be found on the web site or at the course
Blackboard site.
Goals of the Course: In this course, you will:
Do mathematical modeling
which involves studying real world situations using mathematics; in our case,
particularly using ordinary differential equations.
Learn to recognize real world
problems that are or could be examined using mathematical modeling. Be aware of
the power but also the potential weaknesses in using mathematical
modeling.
Develop an understanding of
linear and nonlinear systems and how feedback effects in nonlinear systems
can lead to unexpected behaviors.
Examine differential
equations using graphical (qualitative), numerical and analytical methods.
Communicate your mathematical
reasoning in writing and verbally.
Develop your ability to work
as an independent and selfsufficient learner:
What
to do when you do not know what to do
How
to take what you have learned in one situation and apply it to a new and
different situation (transfer of knowledge)
Get
comfortable with not knowing the answer immediately
Learn
material we have not covered in class by reading the book and applying this
newly learned information to solve problems.
Become
part of a community of learners who support, encourage and learn from one
another.
We
will cover most of the following sections from Blanchard et al:
Ch.1:
Section 1, 2, 3, 4, 5, 6, 7, 8, 9
Ch.
2: Section 1, 2, 3, 4, 5
Ch.
3: Section 1, 2, 3, 4, 5, 6, 8
Ch.
4: Section 1, 2
Ch.
5: Section 1, 2
Ch.
7: Section 1, 2, 3, 4
Appendix
B.
Plus
additional topics as time and interest permits.
The topics from the last time I taught the
differential equations course are listed at the end of the syllabus to give you
a sense of what we will cover.
Additional Reading: In addition to our text, we will have supplementary
readings that show how the material we are learning in the course relates to
real world issues and there will be homework assignments involving the
reading.
Computer Assignments:
There
will be extensive use of the computer during the course both during class time
(with laptops) and as part of homework assignments. We will use the software that comes with the textbook: Differential
Equations.
You
will be encouraged to "play around" with the modules in this program
to develop a graphical understanding of the concepts in our course. We will
also use and write simple programs (possible systems include Excel,
Mathematica). No previous computer experience is necessary or assumed.
Exams:
There
will be a midterm exam, a final exam and a final project. The tentative
schedule for the exams is:
Takehome Midterm
exam: in the 6^{th } (Feb
2125) or 7^{7h} week (Feb
28 March 4).
Final Project: Due at the last class of the semester.
Students
will work in teams on a project of their choosing. The project might involve
using material from the course to study an applied situation, examining a
theoretical issue in more depth or studying a topic that extends the material
from the course. Projects will be written up in the form of a paper (10  15
pages).
Homework:
Homework
will be assigned for each class and will be collected once a week on
Wednesday. Late work will not be accepted unless there is a special situation
(ex. serious medical problem) and you get my permission ahead of time.
The
best way to learn mathematics is by doing lots of problems. Do not limit
yourself to just doing the problems that you are required to hand in. You
should do some problems after each class. This way, the next lecture will make
a lot more sense. Do not wait till the last minute and do all the problems at
once. You will have much more trouble understanding the lectures and will
therefore be using your time inefficiently.
You are encouraged to collaborate on
homework. This means you can talk with each other, figure things out together,
help each other. But when you finally write up your answer, you should do this
by yourself. You may not copy the answer that someone else has written. This
would be a violation of the honor code.
Quizzes:
There
will be a short quiz each week to give you feedback on your progress. Once you
get the quiz back, you are welcome to correct any mistakes and resubmit the
quiz to be regraded.
Classroom:
During
class, there will be a mixture of lecturing by the professor and time spent by
the students working out problems and discussing their results in groups.
Research has shown that this type of active participation leads to improved
learning.
The
group work does not go well when members of the group are absent. Therefore it
is important that you attend to class. Please be respectful of your fellow
students.
If
you decide to take this course, you must commit to attending class regularly.
Attendance will be taken and substandard attendance will be taken into account
in determining grades.
Special Event:
The class will
carry out a roleplaying simulation game that applies aspects of differential
equations to a real world situation. This will require a threehour time block
and will take place during an afternoon or evening in the third or fourth week
of the semester. More details will be forthcoming.
Final grades will be
determined using the following percentages:
Homework, quizzes, class
participation 
25% 
Midterm 
25% 
Final Project 
25% 
Final Exam 
25% 
Total 
100% 
Topics Covered in my Previous
Offering of Differential Equations:
Wk 1:
Mon Jan 16 (Lect 1): Course mechanics,
introduction, computer demonstration, introduction to modeling. (Sect 1.1).
What it means for a function to be a solution of a differential equation. Guess
and Check method. Individual
worksheet, Group worksheet.
For Wed, fill out the information survey part 1
and part
2.
Wed Jan 18 (L2): Examples of models (worksheet), calculating parameter values in
exponential model, how good is the
exponential population model (data
table), logistic population model and computer graphing.
Wk 2:
Mon Jan 24 (L3): Estimating derivative from a
data table. Slope fields (Sect 1.3). Slope field worksheet pt1, pt2. Long term
behaviour, sensitive dependence,
constant solutions (worksheet).
Wed Jan 26 (L4): Finished with worksheet on
long term behavior, sensitive dependence, butterfly effect and chaos; also
constant solutions, attracting (sink) and repelling (source) equilibrium
solutions.
Wk 3:
Mon Jan 30 (L5): Homework session.
Wed Feb 1 (L6): Phase line (picture/portrait)
(Sect 1.6), Euler’s method (Sect 1.4) with group worksheet.
Wk 4:
Mon Feb 6 (L7): Separation of Variables (Sect
1.2), worksheet. Fishing
game. Article about real world
fish stock collapses on reserve in the library.
Wed Feb 8 (L8): Review of modeling via
exponential model, population
models, start of logistic model with harvesting (Sect. 1.7), Harvesting worksheet, Mixing model worksheet. Readings about environmental issues
handed out: “Biologists sort lessons of Fisheries Collapse” and “Canada to
shield 5 million forest acres” – available in hard copy from Prof.
Donnay.
Wk 5:
Mon Feb 13 (L9): Paper review of harvesting
write up (feedback rubric),
harvesting rewrite
instructions. Handout of answer key to mixing problem (paper copy available).
Strategies for
solving modeling problems.
Wed Feb 15 (L10): People who had studied the
same C value compare their results and write up summary. Creating bifurcation
diagram (Sect 1.7) and its implications for fishing limits (group worksheet).
Wk 6:
Mon Feb 20: (L11): Review for midterm (worksheet).
Wed Feb 22: (L12) Start existence and uniqueness of
solutions of differential equations (Sect 1.5). Learn about the harvesting problem of the Tibetan snow lotus
plant and the attempt to determine a sustainable harvesting level on National
Public Radio.
Wk 7: Mon
Feb 27 (L13): Formal Existence Theorem and Uniqueness (notes Sect 1.5), Linearity and
application to linear differential equations (notes Sect 1.8)
Wed
March 1 (L14): Techniques for finding particular solutions to nonhomogeneous
linear differential equations (Sect 1.8). Method of Undetermined
Coefficients = educated guessing!
General first order linear system. Constant coefficients and nonconstant
coefficients. (notes)
Wk 8: Mon
March 13^{th} (L15): Systems of Ordinary Differential Equations (Sect.
2.1, 2.2). Drawing solution
curves using HPGSystemSolver . Graphing
a vector field (pdf
version).
Wed
March 15^{th} (L16): PredatorPrey model derivation. Very simple,
uncoupled system; its solutions and phase space. More complicated model with
coupling caused by predatorprey interaction. Finding equilibrium solutions.
Saddle equilibrium.
Wk 9: Mon
March 20 (L17): Comments about
upcoming hw assignment and how to interpret the coefficients in a model.
Curve
given by parametric equations (x(t), y(t)); its tangent/velocity vector (x’(t),
y’(t)). A curve is a solution of the system of differential equations if its
tangent/velocity vector matches up with the vector field given by the system.
A
nonlinear system can be studied by finding equilibrium points and examining phase
portrait in neighborhood of equilibrium point. These phase portraits agree with
the phase portrait given by a linear system (the linear approximation to the
system, where the approximation is based at the equilibrium point). The
nonlinear phase portrait is created by patching together the linear
approximation pieces. For a while
we will focus on linear systems; such systems can be written in vector notation
using matrices. Interpret a matrix as giving a linear transformation from R^{2}
> R^{2}. Worksheet and graph paper.
Wed
March 22 (L18): Go over the matrix
transformation worksheet. Learn about eigenvectors and eigenvalues (worksheet) (Sect 3.2) and Linearity
(Sect 3.1).
Wk 10:
Mon March 27 (L19): Connect eigenvalues and eigenvectors to systems of
differential equations. Eigenvectors and eigenvalues combine to give particular
solutions – the straight line solutions Y_{1} and Y_{2}. Then using linearity of the system,
which is based upon a matrix transformation being linear, we showed that the
general solution is a linear combination of the straight line solutions: Y(t) =
k_{1} Y_{1}(t) + k_{2} Y_{2}(t). Worksheet.
Wed
March 29 (L20): Practice using eigenvalues, eigenvectors to find solutions of
systems and to draw phase plane pictures by doing a long, multistep problem.
Wk 11:
Mon April 3 (L21): Complex eigenvalues; real and imaginary parts of solutions;
spirals. Sect 3.4. Worksheet.
Wed
April 5 (L22): More on complex eigenvalues. Repeated eigenvalues (Sect. 3.5).
Wk 12:
Mon April 10 (L23): Sect 5.1: Nonlinear systems and linearization. Worksheet.
Wed
April 12 (L24): Sect 5.1: Linearization via Jacobian matrix, Improved numerical
methods (Ch 7). Lab instructions, Euler method, Improved Euler method, RungeKutta method.
Wk 13:
Mon April 17^{th}: Improved Euler, RungeKutta.
Wed
April 19^{th}: review for midterm.
Wk 14:
Monday April 24^{th}: Mathematical epidemiology: SIR and SIRS models.
Wed
April 25^{th}: surveys; playing with fun programs: Butterfly effect;
Lorentz equations, Chemical Osciallations,TD Animations.