## MAT 405, Differential Equations Spring Semester, 2004 Syllabus

Credits: 4

Prerequisite: MAT 162

Lecture: in SE 312B

• Monday & Wednesday, 12:20 - 1:15
• Tuesday & Thursday, 11:00 - 12:20

Instructor: Brett van de Sande

Text: Blanchard, Devaney, and Hall, Differential Equations. See also the Boston University ordinary differential equations project.

 Homeworks 40% Test 1 12% Test 2 12% Test 3 12% Final exam 24%

Methods of solution and application of ordinary differential equations.

Here are some general rules for your homework:

• Write your name, the assigment, and the date on the first page.
• In general, any graphs should be drawn on graph paper.
• Solutions should include intermediate steps.  If you just write down the answer, even if it is correct, you will not get full credit.
• Homeworks must be handed in on time.  Late homeworks, at the discretion of the instructor, will incur a substantial penalty.
The final exam will be open-notebook. The main purpose of the final exam is to see if you have taken good class notes and to see how well you understood the various topics presented in lecture.  The main purpose of the midterms, however, is to see if you understand the homeworks.  The midterms may be closed-book, closed-notes.

I encourage students to work together on the homework problems.  It is often helpful to discuss with others how a problem should be solved. However, when you write down the solution to be handed in, it should be in your own words.  Don't hand in something that you have copied or that you do not understand.

• Homework 1 (PDF)
• Homework 2, due Tuesday, January 27, in class.
Textbook, Section 1.1: 2, 8, 9, 12, 13, 14.
• Homework 3, due Wednesday, January 28, in class.
Textbook, Section 1.2: 1, 9, 10, 20, 21, 22, 27, 29, 30, 35.
• Homework 4, due Friday, January 30, in my office by 4:30 PM.
Textbook, Section 1.3: 3-6, 12 a and b, 15-18.
Use graph paper for any graphs that you draw by hand. For problems 3-6, ignore the directions in the textbook.  Instead:
• Plot the slope field in the region -2 < x < 2, -2 < y < 2 using the technology of your choice.
• Print out your plot.  Your graph should fill a page.
• Based on the slope fields, draw solutions of the differential equation corresponding to boundary conditions y(0)=-1, y(0)=0, y(0)=1, and y(0)=2 on your graph. Include both positive and negative t.
• Homework 5, due Tuesday, February 3, in class.
You must use a spreadsheet to perform the calculations.  If you prefer, you may E-mail me a spreadsheet containing your answers. (Multiple sheets are OK, but multiple files are not.)
Textbook, Section 1.4: 1-11.  For problems 1-4, you must use the step size indicated as well as a step size one 10th as big; plot both answers.
• Homework 6, due Tuesday, February 10, in class.
Textbook, Section 1.5: 5-8, 10a-c, 12, 15-18.
Textbook, Section 1.6: 1-6, 13-18, 30-35, 43, 44-47.
Textbook, Section 1.7: 7-10.
• Homework 7, due Tuesday, February 24, in class. Start by reading Section 2.1 of the textbook.
Textbook, Section 1.8: 7-10 (do by hand), 11-16 (may use technology)
Textbook, Section 2.1: 1-15.
• Homework 8, due Tuesday, March 2, in class.  Use graph paper for your graphs.  Unless instructed to graph by hand, use the technology of your choice for the direction fields. Textbook, Section 2.2: 7-11, 13-16, 17, 19, 21, 23-26, 30.
Textbook, Section 2.3: 1, 2, 6, 7, 9, 10, 19a-c.  Also, plot the solution to 19c in the phase plane.
• Homework 9, due Tuesday, March 9, in class.  Euler's method must be performed with a spreadsheet.  If you prefer, you may E-mail me a spreadsheet containing your answers. (Multiple sheets are OK, but multiple files are not.)
Textbook, Section 2.4: 2, 7, 8, 9, 11, 15.
For problem 11, the exact solution gives y(10)=-0.185346 and v(10)=-0.299783.  What step size do I need for a 10% error?  What about a 2% error?  Graph your solutions in phase space.  What happens to this solution in the infinite t limit?
• Homework 10, due Tuesday, March 23, in class.  Michael Cross at Caltech has some nice material on chaotic systems, including a series of demonstrations of the Lorenz model.
Textbook, Section 2.5: 1, 2.  Also:
• Using the demonstrations, determine how long it takes for two nearby solutions to become completely separate; try initial separations of 0.01, 0.1, 1.0.  What do you conclude about the separation as a function of time?
• If you look at the dependent variables as a function of t, you see that there are two "kinds" of oscillations, oscillations around an equilibrium, and oscillations between the two equilibria.  Roughly, what is the period for each kind of oscillation?
Textbook, Section 3.1: 6, 7, 8, 11, 12, 16, 17, and 19.
• Homework 11, due Tuesday, April 6, in class.
Textbook, Section 3.1: 31, 32, and 33.
Textbook, Section 3.2: 3, 12-20.  For problems 12-14, also plot the direction field (using the technology of your choice) and sketch/plot your solutions on the graph.
• Homework 12, due Thursday, April 8, at 4:30 PM.
Textbook, Section 3.3: 19-21, 23-27.
Textbook, Section 3.4: 1-6.
• Homework 13, due Friday, April 23, at 4:30 PM.
Textbook, Section 3.4: 9, 10, 15.
Textbook, Section 3.5: 1, 2, 5, 6, 12, 17, 18.
• Homework 14, due Tuesday, April 27, in class.
Textbook, Section 3.6: 15, 16, 23, 24, 32, 38.
Textbook, Section 3.8: 3, 7, 16, 17.  Do eigenvalues and vectors on a computer/calculator.
• Homework 15, due Friday, April 30, at 4:30 PM.
Textbook, Section 6.1: 1-8, 17-20.
Textbook, Section 5.1: 3, 13-17.  (for 13-16, look at x >0, y >0.)

Other differential equations courses:
• Oberlin, textbook: Blanchard, Devaney, and Hall, Differential Equations
• Swarthmore, textbook: Blanchard, Devaney, and Hall, Differential Equations (used to be Boyce and DiPrima)
• Stetson, textbook: Blanchard, Davaney & Hall, Differential Equations
• St. Joe's, textbook: Blanchard, Devaney, and Hall, Differential Equations
• Haverford, textbook: Blanchard, Devaney and Hall, Differential Equations
• Colorado State, textbook: Boyce and DiPrima, Elementary Differential Equations and Boundary Value Problems
• Ohio State, textbook: Boyce and DiPrima, Elementary Differential Equations and Boundary Value Problems
• Eastern U., textbook: Boyce and DiPrima, Elementary Differential Equations and Boundary Value Problems
• U. Maryland, textbook: Boyce and DiPrima, Elementary Differential Equations and Boundary Value Problems
• RIT, textbook: Nagle, Saff, Snider, Fundamentals of Differential Equations and Boundary Value Problems
• Calvin College, textbook: Boyce and DiPrima.
• Oklahoma U., textbook: Edwards and Penny, Differential Equations and Boundary Value Problems
• UIUC, textbook: Edwards and Penny, Differential Equations and Boundary Value Problems
• North Dakota, textbook: Kent Nagle and Edward B. Saff, Fundamentals of Differential Equations and Boundary Value Problems
• McGill, textbook: Nagle, Saff, Snider, Fundamentals of Differential Equations and Boundary Value Problems
• Cedarville, textbook: Zill, Differential Equations with modeling applications
• Messiah, textbook: Dennis G. Zill, A First Course in Differential Equations
A list of general resources from MathArchives. The textbooks are all the standard price \$120, used \$50.  Whatever the market will bear...