Schedule, Homework and Project for Ma 341-004 (Fall 2007)

Schedule of Lectures

 This schedule will be roughly followed, I may change it due to questions. It is always good to ask if you don't understand. If you'd like to see another in-between algebra step I'm happy to supply it if you ask.

August

23 - Introduction to Complex Variables, Chapter 1 & 2.2-2.4
28 - Sections 3.1, 3.2, 3.3, 3.4
30 - Sections 4.2, 4.3, & 6.2

September

4 - Sections 4.4, 4.5, & 6.3
6 - Sections 6.2, 6.3
11 - Sections 4.6
13 - Sections 6.4 and review
18 - Test I (n-th order ODEs, 2.1-2.4, 3.1-3.4, 4.2-4.6, 6.1-6.4 and Lecture Notes 1-39 & Complex Notes pages i-8i)
20 - [Project] Section 5.4, Systems and the Phase Plane
25 - [Project] Sections 12.2 & 12.3, Linear and Almost Linear Systems and Stablility
27 - Section 9.1-9.3

October

2 - Section 9.4
4 - Section 9.5
9 - Section 9.6
11 - no lecture - Fall Break
16 - Section 9.7
18 - Section 9.8
23 - Section 9.8
25 - [Project] Energy Analysis, Section 12.4
27 - [Project] Energy Analysis and review

November

1 - Test II (systems of ODEs, 9.1-9.8 and Lecture Notes 72-123)
6 - Section 7.1 & 7.2 & 7.3
8 - Section 7.4
13 - Section 7.5
15 - Section 7.6
20 - Sections 7.7
22 - no lecture - Thanksgiving
27 - Sections 7.8 and review
29 - Test III (Laplace Transforms, 7.1-7.8 and Lecture Notes 40-71)

December

4 - Discussion of Project
5 - Project Due before 4pm in drop-off box.
6 - Review for final
18 - Final Exam (1 - 4pm, same room as usual, it covers the whole course including the Project)



Homework Assigned

 It is important to both complete and understand the homework. I encourage you to form study groups, however, it is very important that in the end you come to an understanding of the material for yourself. You will most likely find the homework in this course challenging at times, so it is important to begin early and give yourself a chance to talk to others (for example me) before it's to late (test time).

  • I do not collect all of the homework, however not doing it would be unwise.
  • Select problems will be collected twice this semester. You will have one week's warning.

    • I am always happy to look over your derivations of homework during office hours, however it is unlikely we will have time to answer all homework questions during lecture. I expect you to complete the homework below. The collected homework makes up 10% of your grade directly, but in practice completing and understanding the homework is one of the keys to mastering this course.

    Chapter 1
    Section 1: 1,3,9,11
    Section 2: 1,3,5,9,11,21,23,30
    Section 3: Play with Direction Fields in maple
    Section 4: Play with the Euler's method in Maple work sheet
    Chapter 2
    Section 1: Read the Book
    Section 2: 1-4, 7-12, 20-26, 29, 31
    Section 3: 1-12, 17-22, 29
    Section 4: 1-8, 13-18, 21-23, 32, 33
    Chapter 3
    Section 1: Read the Book
    Section 2: 1-3
    Section 3: 1-4
    Section 4: 1, 2, 5-8, 25
    Chapter 4
    Section 4.1: Read the Book
    Section 4.2: 1-20, 27-32, 34
    Section 4.3: 1-27
    Section 4.4: 9-12, 14-16, 33, 36
    Section 4.5: 1, 17-27
    Section 4.6: 1-18
    Section 4.8: Read the Book
    Section 4.9: Read the Book
    Chapter 6
    Section 6.1: 8, 9, 11, 14-18
    Section 6.2: 1, 2, 6, 13-15, 18
    Section 6.3: 9, 10, 12, 14, 16, 17, 18
    Section 6.4: 2, 5
    Chapter 7
    Section 7.1: Read the Book
    Section 7.2: 1-4, 10-12, 17-20
    Section 7.3: 1-20, 22
    Section 7.4: 1-3, 7-9, 21-26, 28, 31-36
    Section 7.5: 1-14, 35, 36
    Section 7.6: 5-18, 23-28, 34-37, 58, 59
    Section 7.7: 1-14
    Section 7.8: 1-20, 29
    Section 7.9: 1-6, 17-19
    Chapter 9
    Section 9.1: 11-13
    Section 9.2: 5-13
    Section 9.3: 3-5, 8-14, 17-26, 37-40
    Section 9.4: 3-7, 17-23, 26, 28
    Section 9.5: 1-9, 19-26, 31-33, 41
    Section 9.6: 1, 2, 5-8, 13, 14, 21
    Section 9.7: 1-5, 7, 9, 11-16, 21-23, 31
    Section 9.8: 1, 2, 7-12, 17-24

    The Project
  • due December 5 by 4pm in drop-off box (location TBA)
  • there will likely be four lectures on theory related to this project. Those lectures I have highlighted in blue

    • One important topic that you are not tested on until the final is the phase plane. There are two main goals for this project. The first goal is to initiate you to the concept of the phase plane as a tool to analyze solutions to differential equations. The second goal is for you to use computational tools to aid in the phase plane analysis. The project may be done individually or in a group of upto 3 people. I expect this project will require a nontrivial amount of time and work. You may also compare results with classmates to see if agreement is reached in your analysis. Credit will be rewarded on the basis of clarity of presentation and analysis. Also the group (you may do the project by yourself or in groups of upto 3 people) that is best will earn a bonus point on their overall course average. You should use Maple or Mathematica or Matlab (or some other math software package of your liking) to aid in the creation of the phase planes and other graphs. Do not put the whole thing off until a few days before the due date (December 5), ask me questions earlier so I have time to help you out.

    Because my home computer is quite antiquated by now I ask that you please print out all graphs that are produced through software. I need a hardcopy (paper) to grade. It is perfectly acceptable to add neatly written hand drawn notes to explain what the graphs mean. Your graphs should be labeled so that I can tell what the graph means. About half of the credit in this project is given to correctly and neatly producing the phase plane plots. I think "pplane" in Matlab will do this with minimal effort. There are ways in Maple as well, but I think Matlab has a more robust pre-built package for more or less doing what I ask in the problems below. Notice that I have allowed you to work in groups of three, I would recommend making sure at least one of your group is comfortable tweaking software. If not, perhaps you should regroup.



  • PROBLEM ONE:
    Consider the system dx/dt = x + 2y and dy/dt = 5x - 2y. Complete the following tasks.
    (a.) Use software to plot the phase plane diagram for this system.
    (b.) Classify any critical point(s), that is comment on their type and stability.
    (c.) Solve this system directly using methods from chapter 9.
    (d.) Plot the solutions from part (c.), choose initial conditions so you can compare with (a.)
    (e.) Which uncoupled system is this system is related to. (examples E10, E11, E12, E13 in my notes are examples of uncoupled systems, the x-equation does not involve y and vice-versa). What is the graphical significance of the eigenvectors found in (c.)?


  • PROBLEM TWO:
    Use software to plot the phase plane for the almost linear system
    dx/dt = x + 5y - y^2
    dy/dt = -x - y - y^2
    use algebra to find all the critical points. Classify their type and stability.

  • PROBLEM THREE:
    Consider the system x'' + 2x^2 + x - 1 = 0.
    (a.) Find an equivalent system of first order ODEs in x and v.
    (b.) Use software to plot the Poincare (xv-plane) phase plane diagram.
    (c.) Use the energy analysis to sketch the phase plane plot for this conservative system.
    (d.) Comment on the possible motions of the system, are there any unbounded trajectories, closed orbits or other interesting features?

  • PROBLEM FOUR:
    Consider the system x'' + 2x^2 + x - 1 + x' = 0.
    (a.) Find an equivalent system of first order ODEs in x and v.
    (b.) Use software to plot the Poincare (xv-plane) phase plane diagram.
    (c.) Use the energy analysis to sketch the phase plane plot for this nonconservative system.
    (d.) Comment on the possible motions of the system, are there any unbounded trajectories, closed orbits or other interesting features? How do these contrast with the analagous trajectories in PROBLEM THREE ?

    Back to MA341 section 4 Home

    Last Updated: 8-17-07