Ordinary Differential Equations

MATH 304-01, Exam #2

Wednesday, Nov. 10, 1:00 - 1:50 pm

The second exam covers the supplementary material from the Blanchard, Devaney, Hall text on harmonic oscillators (Sections 3.6, 4.1 and 4.2) as well as Sections 3.1 - 3.4, 4.1, 5.2, 5.5, 6.1 and 6.3 of the course text. In some cases, your class notes will be more useful than the text as we covered certain topics in greater detail in class. Also note that the material from the first exam should not be forgotten. For example, to draw a phase portrait of a real sink, you need to find the eigenvalues and eigenvectors (essentially covered in Chapter 2.) You should go over homework problems and your class notes. The exam will be designed to take 45-50 minutes.

Note: You will be given a scientific calculator for the exam which does NOT have graphing capabilities so be prepared to answer questions without your personal calculator or computer. The only numerical computations asked will be doable on a basic scientific calculator.

Exam Review: We will review for the exam on Monday, Nov. 8th from 6:00 - 7:15 pm in Swords 359. Please come prepared with specific questions.

The following concepts, definitions and topics are important material for the exam:

  1. Linear Algebra: linear independence, basis, trace, determinant, eigenvalues and eigenvectors, characteristic polynomial, null space, canonical or Jordan form
  2. Planar Linear Systems -- Real, distinct eigenvalues: straight-line solutions, phase plane diagrams, form of general solution, stability type (source, sink, saddle, degenerate)
  3. Planar Linear Systems -- Complex eigenvalues: phase plane diagrams, form of general solution, stability type (spiral source, spiral sink, center)
  4. Planar Linear Systems -- Repeated eigenvalues: phase plane diagrams, form of general solution, stability type (repeated source, repeated sink, degenerate)
  5. Planar Linear Systems -- Classification: Jordan form, finding the change of coordinates matrix P, generalized eigenvectors, linearity principle, trace-determinant plane
  6. Harmonic Oscillators: physical setup (parameters m, b and k), converting to a system, solving homogeneous and non-homogeneous cases, classification (overdamped, underdamped, critically damped, undamped), long-term behavior, describing the motion, phase plane diagrams and component graphs, complexification
  7. Higher Dimensional Linear Systems: Jordan form, finding the change of coordinates matrix P, generalized eigenvectors, solving X' = A X, understanding phase portraits

Some Practice Problems:
(The answers to the odds for Blanchard, Devaney and Hall are in the back of the book on reserve.)

Chapter 3 Exercises (pp. 57 - 60)
Problems:   4, 5, 7, 8, 16

Chapter 5 Exercises (pp. 104 - 106)
Problems:   2d, 5b, 5e, 5g

Chapter 6 Exercises (pp. 135 - 138)
Problems:   1b, 1g, 6

Blanchard, Devaney, Hall handouts
Section 3.6 :   13, 18, 19, 21, 26, 27, 32, 38
Section 4.1 :   7, 11, 15, 33, 37
Section 4.2 :   5, 7, 11, 20, 21, 23

Additional Problems:
1. Given the following one parameter family of linear systems X' = AX, sketch the path traced out by the family in the trace-determinant plane. Describe the bifurcations that occur along this path and compute the corresponding values of a.

A =   a   1
  a   a


2. In the ab-plane, identify all regions where the following linear system has the same stability types (ie. saddle, sink, source, etc.)

A =   a     b
  -b   a


3. Consider a harmonic oscillator with mass m = 1, damping coefficient b = 2 and arbitrary spring constant k. Describe the different types of oscillators as k varies. Where do bifurcation(s) occur? Describe the behavior before and after.

4. Suppose that A is a 4 x 4 matrix with repeated eigenvalue 2. Describe all possible canonical forms for A.

5. Give an example of a 3 x 3 matrix A for which solutions to X' = A X that begin in the xz-plane spiral towards the origin but solutions starting anywhere else head away from the origin. How do solutions not in the xz-plane behave?