Ordinary Differential Equations

MATH 304, Exam #2

Thursday, Nov. 20, 6:30 - 8:00 pm, Swords 359

The second exam includes all of the material we have covered in Chapters 2 and 3. This is Sections 2.1 - 2.6 and 3.1 - 3.7, homework assignments 5 - 8, and Lab 2. You should primarily focus on homework problems and your class notes. I have also chosen some good practice problems from the Chapter 2 and 3 reviews, listed below. The answers to all of the review problems are posted on the Moodle page for the course. The exam will be designed to take one hour although you will have the full 90 minutes.

Note: You will be allowed a scientific calculator for the exam that does NOT have graphing capabilities. Please bring your own scientific calculator.

Exam Review: We will review for the exam during Wednesday's class on Nov. 19. Please come prepared with specific questions.

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

  1. First-order Systems: equilibrium points, phase portraits, vector field, direction field, predator-prey system, sketching component graphs from the phase portrait, Euler's method, Existence and Uniqueness Theorem

  2. Linear Algebra: linear independence, basis, trace and determinant of a matrix, eigenvalues and eigenvectors, characteristic polynomial, null space of a matrix

  3. Planar Linear Systems: linearity principle, eigenvalues and eigenvectors, stability type (source, sink, saddle, spiral sink, center, etc.), straight-line solutions, sketching phase portraits, sketching component graphs, finding the general solution for three cases (real distinct, complex, or repeated eigenvalues), solving an initial-value problem, trace-determinant plane, bifurcations

  4. Harmonic Oscillators: physical setup (parameters m, b, and k), converting to a system, classification (overdamped, underdamped, critically damped, undamped), finding the general solution, finding a specific solution given an initial condition, long-term behavior, describing the motion of the system, sketching phase portraits and component graphs

Chapter 2 Review Exercises (pp. 224 - 228)
Problems:   1, 2, 3, 5, 7, 9, 12, 13, 15, 16, 20, 23, 24, 25, 26, 27, 28, 31, 32, 33, 34, 36

Chapter 3 Review Exercises (pp. 376 - 380)
Problems:   1, 2, 3, 4, 5, 7, 9, 10, 11, 12, 13, 15, 19, 20, 21, 23, 24, 25, 26, 27, 28, 30, 31, 32

Additional Problems:
1. Given the following one parameter family of linear systems Y' = AY, 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. Consider a harmonic oscillator with mass m = 1, damping coefficient b = 2, and arbitrary spring constant k (assume k > 0). Describe the different types of oscillators as k varies. Where do bifurcation(s) occur? Describe the behavior before, at, and after any bifurcations.