This page gives an approximate schedule of the topics covered in lecture and related homework problems.

M Aug 18 Class business;
         What is an ODE? 
         What will we learn in this course (outline)

W  20    Modeling via Calculus of Variations (intro)
         First Assignment

M  25    Some basic things:
         Numerical solutions
         Local existence and uniqueness
         Second Assignment         
         (Actually, we didn't do any of this stuff.)
         What we really did is continue our discussion of minimization in 
         the Calculus of Variations (including some basic notation from 
         analysis (continuity classes) and topology (open and closed sets)).
         Also, I told Melanie to look at Suppes' "Axiomatic Set Theory" for 
         a discussion of cardinality, but that was a mistake.  What I really 
         meant was "Naive Set Theory" by P. Halmos.  

W  27    (moving stuff down from above)
         Some basic things:
         Numerical solutions
         Local existence and uniqueness (didn't get to this)

M Sept 1 holiday

W   3    Existence and Uniqueness (local)
         Third Assignment         
         Interval of existence (discussed)
         Slope Fields (not done)
         Other things to do when you can't solve an ODE 
         "Qualitative methods" (still to come)

M   8    Finished the proof of existence via successive approximations.
         slope fields
         Worked out an Euler-Lagrange Equation (Mel and Sean)

W  10    Review\summary of existence/uniqueness
         linear equations
         why approximate functions with smooth ones
         constraints in calculus of variations

Th 11     Practice Exam 

M  15    A little more on hyperbolic conservation laws and pde

W  17    Modeling motion via Hamilton's Action Principle
         The hanging chain and problems with constraints

M  22    General Theory of Linear ODE (relation to systems)
          Fourth Assignment 
W  24    Systems and Dynamical Systems
Th 25    Midterm Exam

M  29    Systems---Dynamical Systems (introduction)

W Oct 1  Systems (continued) Stability of Equilibria  
         Proust lecture

M   6    The beginning of 2x2 linear constant coefficient systems
         Beware:  Sean made the observation (and I agreed) that one can 
         look at the eigenvectors and eigenvalues and write down the 
         solutions and draw the phase diagram.  This is sort of *half* true.
         (It was nice to indulge in that illusion for a moment!)  But that 
         "big picture" relies on the assumption that your matrix *has* a 
         basis of (real) eigenvectors.  Outside of borderline cases, there 
         is the possibility that you get complex eigenvalues and eigenvectors.
         This is the other half of the story, and you can start to think 
         about it with the system x' = -y, y'=x.

W   8    Fitting together the phase plane diagram for a nonlinear system
         Checklist for nonlinear systems:
         * Existence/Uniqueness
         * isoclines
         * linearization (constant coeff. systems)
         * Hamiltonians (conserved quantities)
         * Liapunov Functions
         * Numerical solutions
         * 1D dynamics
         * 2D dynamics (Poincare-Bendixon, Omega limit sets)
         * 3D dynamics (Lorenz Attractor)
         * Bifurcation Theory
         * Chaos
          Fifth Assignment 

M  13    2nd order equations/springs and things
         (relation of 2nd order equations to Newton's Law)

W  15    More on 2nd order equations
         Forcing and Resonance (outline)

M  20    Relation of 2nd order equations to calculus of variations
         Hanging Chain (easy project)
         Exotic Tubes (hard project)


W  22    More on the hanging chain; The first integral

M  27    PDE from the calculus of variations; Capillary surfaces

W  29    Exotic Tubes

          Sixth Assignment 

M Nov 3  Laplace Transforms 

W   5    Laplace Transforms (continued)

M  10    Fourier Series Solutions

W  12    Example x''+9x = periodic step function (continued)

          Seventh Assignment 

M  17    Example (continued)
         Hamiltonian systems

W  19    PDE, separation of variables, and
         Sturm-Liouville Theory

M  24    Series Solutions and Sturm-Liouville Theory
         Project presentations?
W  26    

M Dec 1  

W   3

         Fourier Series
         Laplace Transforms
         Projects
         PDE
         Don't forget checklist for nonlinear ODE above.