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SOLVING SECOND ORDER DIFFERENTIAL EQUATION NON-LINEAR
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ORDINARY DIFFERRENTIAL EQUATIONS, SPRING 2006
An introduction to elementary techniques, concepts, theory and applications.


SHORTCUT TO THE ASSIGNMENTS:
    Assignment: | |
ANNOUNCEMENTS:
  • The final exam is now posted. I selected a due date after the break. If you want to wait to take it, I will have office hours devoted to ODEs when you get back, to ask any last-minute questions.
  • My office hours generally will be Thursday, 4:00-5:30 and Friday, 4:00-5:30. In addition, I will schedule evening hours once or twice each week.
  • Unless otherwise indicated, homework will be dueFridays at 5:30 PM inthe "In Box" outside my office, Olin Center 261. Graded homework papers will be available in a box outside my office in the Olin Center.
0. Information
    Outline -- The course outline is available here.
    Reading -- Read the note on written homework found here.
    Problem -- If you have not filled one out for me before, please complete the information sheet found here and bring it to class on Tuesday, January 31.
Final Examination Due Thursday, March 30.You can download the final examination when you are ready totake it. Budget three hours. You may download the instructions for thefinal at any time.The exam is thorough, and substantially based on problems verysimilar to homework exercises and quizzes. You can best prepare byreviewing the topics that were covered by the quizzes, by reading thetextbook examples in the sections that we covered, and by reviewingyour homework notebooks. There are 12 problems on the exam. The first8 are largely mechanical and cover all of the basic methods forsolving first and second order equations that we have learned. Thelast four problems are application problems that involve analyzing agiven model, or setting up a simple ODE from a description and thenanalyzing that. I would say that of the problems, two of them aresomewhat challenging, seven of them are eight of them arestraightforward, and the other two somewhere in between. You may takethe exam at any time. If you want to finish it before break, turn itin at my office. If you want to finish it over break, you may fax itto me at 781-292-2505.

Here are some good practice problems for the exam, from the text:2.1.10, 2.1.11, 2.2.8, 2.2.20, 2.3.4, 2.5.17, 2.5.23, 3.1.10, 3.4.18,3.4.12, 3.6.6, 3.6.9, 3.6.13, 3.7.2, 3.7.3, 3.8.27, 3.9.11.

6. Due Wednesday, March 15 : Applications of second-order equations to mechanical systems; higher order constant coefficient equations; linear systems of differential equations.
    Read these sections in the text: 3.9; 4.2; 7.5

    Write-up the following graded exercises. Be sure to read the guidelines for writing homework assignments.

    (3.9) 9, 12, 13, 17
    (4.2) 39
    (7.5) 1, 5 (We will cover this topic in class on Tuesday.)

Quiz 3. Due Tuesday, March 13 . This quiz focuses on solving second-order differential equations and applications to mechanical systems. Key topics to review: solutions to second order homogeneous differential equations; linear independence and dependence of solutions; constant coefficient equations; initial value problems; method of undetermined coefficients for non-homogeneous equations; mechanical spring-mass systems.

You can download Quiz 3 when you are ready to take it. The quiz will take 40 minutes. You will be asked on the quiz to use your computer to produce graphs of solution formulas that you detemine. Before starting the quiz, please review how to produce graphs of functions in Matlab or Maple. You might review how to draw a graph of a function that is piecewise defined (e.g. y = f(t) for 0
5. Due Wednesday, March 8 : Applications of second-order equations to mechanical systems; higher order constant coefficient equations.
    Read these sections in the text: 3.8, 3.9, 4.1, 4.2

    Write-up the following graded exercises. Be sure to read the guidelines for writing homework assignments.

    (3.8) 6, 9, 27, 29
    (4.2) 12, 17, 29, 34 (you may use a computer or calculator to find zeros of the characteristic polynomial)

Quiz 2. Due Friday, March 3. This quiz focuses on solving second-order differential equations. Key topics to review: solutions to second order homogeneous differrential equations; general solution; wronskian; linear independence and dependence of solutions; constant coefficient equations; initial value problems; method of undetermined coefficients for non-homogeneous equations.

You can download Quiz 2 when you are ready to take it. The quiz will take 40 minutes.

4. Due Thursday, March 2 : Second order linear equations; initial value problems; fundamental sets of solutions; constant coefficient, homogeneous equations; non-homogeneous equations; method of undetermined coefficients; method of variation of parameters; applications to mechanical systems; resonance; beats.
    Read these sections in the text: 3.6, 3.7, 3.8, 3.8

    Work on the following practice exercises, that you should keep a record of in a notebook, but do not have to write-up and turn in:

    (3.6) 1, 2, 6, 11
    (3.7) 1, 3
    (3.8) 1, 7, 15

    Write-up the following graded exercises. Be sure to read the guidelines for writing homework assignments.

    (3.4) 14, 22, 33
    (3.6) 8, 11, 13, 29
    (3.7) 2, 5, 21
    (3.8) 11, 13, 21

3. Due Thursday, February 23: Second order linear equations; wronskian and linear independence of solutions; initial value problems; existence and uniqueness; fundamental sets of solutions; constant coefficient, homogeneous equations; ODEs as linear operators; Euler's formula; reduction of order.
    Read these sections in the text: 3.1, 3.2, 3.3, 3.4, 3.5

    Work on the following practice exercises, that you should keep a record of in a notebook, but do not have to write-up and turn in:

    (p.132) 34a
    (3.1) 1, 3, 5, 13
    (3.4) 2, 4, 6, 8 (3.5) 1, 5, 7, 9, 17

    Write-up the following graded exercises. Be sure to read the guidelines for writing homework assignments.

    (p. 132) 33, 35
    (3.1) 8, 15, 21
    (3.4) 10, 14, 19, 24, 33 (tricky!)
    (3.5) 10, 14, 18, 19

Quiz 1. Due Monday, February 20. This quiz focuses on solving first-order differential equations. Key topics to review: integrating factor method; existence and uniqueness of solutions; initial value problems; separation of variables; domains of solutions. The quiz is 60 minutes; no notes or use of the text.

You can download Quiz 1 when you are ready to take it.

2. Due Friday, February 10: Linear and non-linear equations; separation of variables; integral curves; initial value problems; physical applications; non-dimensionalization and change of variable; autonomous equations; population models.
    Read these sections in the text: 2.3,2.4; 2.5

    Work on the following practice exercises, that you should keep a record of in a notebook, but do not have to write-up and turn in:

    (2.4) 13, 14, 15, 17, 27
    (2.5) 1, 2, 4, 7, 10, 15

    Write-up the following graded exercises. Be sure to read the guidelines for writing homework assignments.

    (2.3) 25
    (2.4) 25, 26, 28, 32 (Note: These problems do not depend on the Theorems in (2.4) )
    (2.5) 3, 19, 20

    In addition do the following problem: For an object moving vertically in a uniform gravitational field with a viscous drag force proportional to velocity: (1) Write the equation of motion for velocity (2) Determine a change of scales to non-dimensionalize the equation (3) Solve the non-dimensional equation (4) From this solution, determine the solution to the problem in the original units.

1. Due Friday, February 3: Introduction to ODEs; classification of equations; solutions; direction fields; first-order linear equations; integration factor method; initial value problems; non-linear equations; separation of variables; physical applications; existence and uniqueness.
    Read these sections in the text: 1.1, 1.2, 1.3; 2.1, 2.1, 2.3,2.4; 2.5

    Work on the following practice exercises, that you should keep a record of in a notebook, but do not have to write-up and turn in:

    (1.1) 1, 3, 22, 23
    (1.2) 9, 17
    (2.1) 5, 6, 9, 10, 17, 21, 22
    (2.2) 1, 4, 15, 21, 27
    (2.3) 1, 20, 21

    Write-up the following graded exercises. Be sure to read the guidelines for writing homework assignments.

    (2.1) 31, 38
    (2.2) 16, 21, 30
    (2.3) 2, 15, 22



1-26-06