Ordinary Differential Equations

MATH 304, Exam #2

Wednesday, Nov. 15, 5:30 - 7:00 pm, Swords 302

The second exam includes all of the material we have covered in Chapters 2, 3 and 4. This is sections 2.1 - 2.4, 3.1 - 3.7, and 4.1 - 4.3 and homework problems sets 5 - 9. You should go over homework problems and your class notes. I have also chosen some good practice problems from the Chapter 2, 3 and 4 reviews. The answers to the odd numbered questions are in the back of the book while the answers to the evens are given below. The exam will be designed to take one hour although you will have the full 90 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 a computer.

Exam Review: We will review for the exam during Monday's class, Nov. 13th. Please come prepared with specific questions. Class on the day of the exam is canceled.

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 phase portrait, Euler's method, Existence and Uniqueness Theorem
  2. Linear Algebra: linear independence, basis, trace, determinant, 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, form of general solution for three cases (real distinct, complex or repeated eigenvalues), trace-determinant plane
  4. Harmonic Oscillators: physical setup (parameters m, b and k), converting to a system, solving homogeneous and non-homogeneous cases, extended linearity principle, classification (overdamped, underdamped, critically damped, undamped), long-term behavior, describing the motion, sketching phase portraits and component graphs, complexification, beats and resonance

Some Practice Problems:

Chapter 2 Exercises (pp. 220 - 223)
Problems:   1, 3, 5, 7, 9, 12, 13, 18, 19, 21, 23, 27, 29, 30, 31
The answers to the evens are:
12.   All solutions move away from the origin as t increases (or head towards the origin in "backwards" time.)
18.   True; a solution to an autonomous systems can have an arbitrary shift in the t-variable and still be a solution.
30.   x(t) will start at 1, decrease until a small negative value, than increase towards 0. y(t) monotonically decreases from 1 to 0.

Chapter 3 Exercises (pp. 370 - 374)
Problems:   1, 3, 5, 7, 9, 10, 11, 12, 15, 17, 19, 20, 21, 23, 25, 27, 29, 31, 32
The answers to the evens are:
10.   x(t) = y_0 t + x_0, y(t) = y_0; phase portrait has a line of equilibria at y=0 and all other solutions lie on lines y = y_0.
12.   True
20.   (a) D = -3 T, (b) bifurcations at a = -12 (repeated sink) and a = 0 (line of equilibria)
32.   (a) Y(t) = k_1 e^(-2t) (1, -2) + k_2 e^(-t) (1, -1), (c) Y(t) = -3e^(-2t) (1, -2) + 3e^(-t) (1, -1).

Chapter 4 Exercises (pp. 443 - 445)
Problems:   1, 2, 3, 7, 8, 13, 15, 19, 21, 23, 25a, 27a
The answers to the evens are:
2.   w = 2
8.   The coffee has a natural sloshing frequency in the cup. Walking down the stairs provides a type of periodic forcing. If the frequency of walking is close to the natural sloshing frequency of the coffee, the amplitude will be high and spilling can occur. This only happens when you are close to resonance.

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. 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 any bifurcations.