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., E-mail: vdonnay1

Office Hours: tba

 

Pre-requisites: You should have taken Calculus of Several Variables/ Multi-variable 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, 3nd 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 non-linear systems and how feedback effects in non-linear 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 self-sufficient 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 mid-term exam, a final exam and a final project. The tentative schedule for the exams is:

 

Take-home Midterm exam: in the 6th  (Feb 21-25) or 77h 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 re-graded.

 

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 role-playing simulation game that applies aspects of differential equations to a real world situation. This will require a three-hour 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 non-homogeneous linear differential equations (Sect 1.8). Method of Undetermined Coefficients  = educated guessing! General first order linear system. Constant coefficients and non-constant coefficients. (notes)

Wk 8: Mon March 13th (L15): Systems of Ordinary Differential Equations (Sect. 2.1, 2.2). Drawing solution curves using HPGSystemSolver . Graphing a vector field (pdf version).

                  Wed March 15th (L16): Predator-Prey model derivation. Very simple, uncoupled system; its solutions and phase space. More complicated model with coupling caused by predator-prey 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 non-linear 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 non-linear 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 R2 -> R2.  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 Y1 and Y2.  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) = k1 Y1(t) + k2 Y2(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, multi-step 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: Non-linear 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, Runge-Kutta method.

 

Wk 13: Mon April 17th: Improved Euler, Runge-Kutta.

                  Wed April 19th: review for midterm.

Wk 14: Monday April 24th: Mathematical epidemiology: SIR and SIRS models.

                  Wed April 25th: surveys; playing with fun programs: Butterfly effect; Lorentz equations, Chemical Osciallations,TD Animations.