WEEK 1.

MODELING: Write down equations 1. for population models,
          2. from Newton's 2nd law, and 
          3. for simple (unconstrained) variational problems 
             (Euler-Lagrange egns.)
EXPLICIT SOLUTIONS: Solve y'=g(x).
ANALYSIS: Prove existence and uniqueness for y'=g(x).
QUALITATIVE: Rough sketches of solutions for y'=f(y).
NUMERICAL: Write a program that implements Euler's method.


WEEK 2.

EXPLICIT SOLUTIONS: Separable equations
ANALYSIS: 
QUALITATIVE: Slope fields;
NUMERICAL: Plot slope fields with a math package


WEEK 3.

EXPLICIT SOLUTIONS: Second order linear constant coeffs.
ANALYSIS: Existence and Uniqueness Thm. 
          (use to analyze single first order eqn.)
          Write any regular ODE as a first order system.
QUALITATIVE: Phase space for a single equation.


WEEK 4.  

(A weak week mathematically due to hijackings etc.)
ANALYSIS: Behavior of solutions for a single autonomous 
          equation x'=f(x) in terms of the zeros of f
Mathematical Culture: The Poincare Conjecture.

WEEK 5.

QUALITATIVE: Flow terminology; Classification of equilibria;
             Bifurcations.
ANALYSIS: Linearization.
MODELING: You should be starting on your projects.

WEEK 6.

QUALITATIVE: Bifurcation diagrams
ANALYSIS: Bifurcation stability theorem
EXPLICIT SOLUTIONS: First order linear equations
MODELING: Mixing problems

WEEK 7.

MODELING: Predator-prey systems; Newton's 2nd law
EXPLICIT SOLUTIONS: Simple first order systems
ANALYSIS: Uniqueness applied to orbits in phase space
NUMERICAL: Euler's method for systems

WEEK 8.

QUALITATIVE: Definitions of limit sets, orbits, etc.
              All possible limit sets in R^1 and R^2
EXPLICIT SOLUTIONS: Singular 2X2 systems
ANALYSIS: Changing basis

WEEK 9.

LINEAR 2X2 SYSTEMS WITH CONSTANT COEFFICIENTS:
ANALYSIS: Forward and reverse asymptotics 
QUALITATIVE: Know all possible phase diagrams;
             Be able to determine from eigenvalues
EXPLICIT SOLUTIONS: Can solve all

WEEK 10.

MODELING: Harmonic oscillators with linear damping
EXPLICIT SOLUTIONS: Nonhomogeneous equations with simple forcing
QUALITATIVE: 3X3 systems

WEEK 11.

MODELING: Recognizing resonance in physical systems
QUALITATIVE: Simple coupling of modes; boundedness
ANALYSIS: Computing the limits associated with resonance

WEEK 12.

ANALYSIS: Solving a forced harmonic oscillator equation using 
          Fourier Series

WEEK 13.

EXPLICIT SOLUTIONS: Solving ODE with Laplace Transforms
MATHEMATICAL THEORY: Mapping spaces of functions (transforms)

WEEK 14.

Nothing new...just practice...and being thankful

WEEK 15.

MODELING: Impulse forcing
EXPLICIT SOLUTIONS: Solving equations with impulse forcing 
                    (using Laplace Transforms)
MATHEMATICAL THEORY: Distributions (Definition, distributional 
                         derivatives and Laplace Transforms)

WEEK 16.

ANALYSIS: Nonlinear Systems: 
          1. How to linearize, use nullclines, etc.
          2. How to use a Hamiltonian.