Schedule, Homework and Project for Ma 341-002 (Summer I 2008)

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.

May

19 - Chapter 1 & 2.2-2.4
19 - Sections 3.1, 3.2, 3.3, 3.4
20 - Introduction to Complex Variables
20 - Sections 4.2, 4.3, & 6.2
21 - Sections 4.4, 4.5, & 6.3
22 - Sections 6.2, 6.3
23 - Sections 4.6
23 - Sections 6.4
23 - Homework I due at end of class, but please don't miss class
26 - No lecture, Holiday.
27 - review
28 - 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)
29 - Section 9.1-9.3
30 - Section 9.4

June

2 - Section 9.5
3 - Section 9.6
4 - Section 9.7
5 - Section 9.8
6 - Section 9.8
6 - Homework II due at end of class, but please don't miss class

9 - [Project] Section 5.4, Systems and the Phase Plane
9 - [Project] Sections 12.2 & 12.3, Linear and Almost Linear Systems and Stablility
9 - [Project] Energy Analysis, Section 12.4
9 - [Project] Energy Analysis and review

10 - Test II (systems of ODEs, 9.1-9.8 and Lecture Notes 72-123)
11 - Section 7.1 & 7.2 & 7.3
11 - Section 7.4
12 - Section 7.5
13 - Section 7.6
16 - Project due at beginning of class
16 - Sections 7.7
17 - Sections 7.8
18 - Discussion of Project
18 - Homework III due at end of class, but please don't miss class
19 - review for test III
20 - Test III (Laplace Transforms, 7.1-7.8 and Lecture Notes 40-71)
22 - Q&A session for final, time and location TBA.
23 - Final Exam (8am - 11am, 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 three times this semester. These are the "Required" homework sets.

    • It is wise to do enough of the textbook homework that you are confident you could do the rest of it. The problem sets linked below are the "Required Homework Sets" (this is perhaps misleading because I don't expect everyone to do all of it, some of the problems should be considered as challenge bonus problems, ask me if in doubt and I'll tell you which ones are more towards the purpose of test preparation). I have made the homework due the day before the review day for each test. This is a little rough, you may have very little time to finish the later sections. My reasoning for this is simple, I would rather you lose a point or two on the homework and have a chance to fix it before the test. My goal is to return the homework to you on the review day. I will have solutions posted shortly after you turn in the homework so late homework is not generally accepted.

  • Required Homework One (upto 5pts of total grade) DUE MAY 23
  • Required Homework Two (upto 5pts of total grade) DUE JUNE 6
  • Required Homework Three (upto 5pts of total grade) DUE JUNE 18

    • 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 understand how 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
    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 June 16 at beginning of class
  • there will a lecture on theory related to this project on June 9. Those materials 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. 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 (June 16), 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 = 3x + 4y. 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 mx'' + kx + sin(wt) = 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 for several choices of m,k,w (positive constants).
    (c.) Let w = 0 (just for this case) 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? How does the answer to this question depend on the relation of k and m to w? (I will award substansial partial credit for analyzing the w=0 case, but I'd like to see what happens with the sin(wt) nonzero.)

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    Last Updated: 5-19-08