This page gives an approximate schedule of the topics covered in lecture and related homework problems.
M Aug 19 Class business;
Calculus of Variations;
What is an ODE?
Exercise: How do you minimize a functional E:F --> R?
Read Section 1.1 and do some exercises.
W 21 Calculus of Variations (cont.)
What is an ODE? (cont.)
Exercises: 1. Think about the area functional. (Should the
(constant multiple of 2 pi matter in finding a minimizer?)
2. Find all solutions of the n-th order equation
y^(n) = p(x)
where p(x) is a polynomial and y = y(x).
3. Find a first order system that is equivalent to the n-th
order ODE y^(n) = (y^2+7)log(y').
F 23 Playing around with the area of
rotationally symmetric surfaces.
Exercises: 1. Compute A(f) for various functions (e.g., when
the graph of f is a straight line, a piece of a parabola, or
an arc of a circle.) What kind of function gives the least
area?
2. Prove that if the minimizer has any point on the axis of
rotation, then the minimizer must be the singular solution
given in class (two disks).
3. Compute the potential energy of a chain of constant linear
density rho whose position is given by the graph of a function
f with prescribed values at the endpoints of an interval.
4. Compute the time it takes a bead that starts from rest at
(0,c) and slides without friction under the force of gravity
down a straight rod to the point (b,0).
M 26 The First Variation (Part I)
ODE Overview
Exercises: 1. If integral(h(x) eta(x)) dx = 0 for all bumps
eta, what can you say about h?
2. (Book problems (first edition) 1.2.1,2,4)
3. (Book problem (first edition) 2.3.13(a); write down the
associated system)
W 28 The First Variation (Part II)
F 30 Variational ODE (Euler Lagrange Equation)
Read section 1.2, work some problems
M Sept 2 Holiday
W 4 Calculus of Variations (examples)
Separable Equations (Section 1.2 in the book)
Exercises: 1. Find the first integral for the minimizer
of A(y) = int y sqrt(1+y'^2) dx (the area integral).
Solve the resulting first order ODE.
2. Minimize E(y) = int sqrt(1+y'^2) dx over C^1 with
fixed endpoints. What is E(y) geometrically?
Th 5 Separable Equations
(Final comments on the overview of ODEs---the big picture)
F 6 Quiz 1 Calculus of variations
M 9 Modeling Populations
Exercise Do some mixing problems like 1.2.40.
W 11 Direction Fields and Vector Fields
Exercises: 1. Read section 1.3 and work some problems.
2. Use Mathematica to plot the direction field of a first
order equation (try one automonomous, one of the form
y'=f(x), and one that is niether of those). Use
Mathematica to plot the vector field of a 2x2 system.
F 13 Dynamical Systems Terminology for Autonomous Systems
Quiz 1 reworks due
M 16 1.8 Linear First Order Equations
(plus some comments on general linear equations
and existence and uniqueness)
Do some problems from section 1.8
W 18 3.6 Second Order Linear Equations with constant coeffs.
F 20 Class will be held in Skiles 202.
M 23 More remarks on linear equations
1.4 and 2.4 Numerical Methods
Problems 1.4.5,6,12
W 25 Quiz 2 First Order Equations
F 27 1.5 Existence and Uniqueness (a second pass)
(Long time existence for linear equations)
Problems 1.5.12,13,18
M 30 1.6 First Order Autonomous Equations; Linearization
Exercises: 1. Consider the equation y'=f(y) with f Lipschitz.
Assume y_1 is the greatest zero of f and y_0 > y_1. Show
that if y is a solution with y(0)=y_0 and y is defined for
0 < t < infty, then (a) y is defined for all time, and (b)
lim(as t --> infty) y(t) = infty.
2. Prove that in one dimension, attractive implies strictly stable.
Problems: 1.6.1-12,29-36
T Oct 1 First Order Autonomous Equations (cont.)
W 2 Linearization of a single equation
Th 3 Quiz 3 Classifying Equations; Linear Const. Coeff. Equations
F 4 RM 202 Skiles
Chapter 3
2x2 Linear Systems with Constant Coeffs. (Linear Equivalence)
M 7 Back in RM 246 Skiles
The First Variation (Part III) Lagrange Multiplier)
W 9 RM 202 Skiles
Analysis of Canonical Systems
F 11 RM 202 Skiles
Matrix Exponentials
M 14 Fall Break
W 16 RM 202 Skiles
Linearization of Plane Systems
F 18 RM 202 Skiles
Linearization of Plane Systems (cont.)
M 21 Review
T 22 Exam 4 (serious exam)
Existence/Uniqueness, Dynamical Systems Terminology,
Single First Order Autonomous Equation
W 23 Harmonic Motion; Second order nonautonomous equations
Relevant Sections: 2.1-2; 3.1-6 (work problems from these)
Th 24 4.1-2 Resonace and Forcing
F 25 Resonance and Forcing (cont.)
M 28 4.2-4 Resonance and Near Resonance
Problem 4.3.26
Work Problems in 4.4
W 30 5.1 Linearization
F Nov 1 5.1 Linearization (cont.)
Work Problems in 5.1
M 4 5.2 Global aspects of nonlinear systems (nullclines;
separatrices)
Work Problems in 5.2
W 6 5.3-4 Hamiltonian Systems and Liapunov Functions
Work Problems in 5.3-4
5.6 Chaos
F 8 Exam 5 Linear Constant Coefficient Systems and Linearization
M 11 More Chaos
T 12 Laplace Transforms (Chapter 6)
W 13 1.7 Bifurcations
Th 14 Laplace Transforms
F 15 Problems on Laplace Transforms
Numerics and numerical packages
M 18
T 19 Shifting along the frequency axis
W 20
Th 21 Shifting along the time axis
F 22 Second project checkpoint; preliminary webpages due
M 25 Exam 6 (tentative date)
Forcing and Resonance, Nonlinear Equations (global aspects),
Computing Laplace Transforms
T 26 Convolution and review
W 27 Distributions (preliminaries; impulsive forces)
Th 28-29 Holiday
M Dec 2 Distributions (part II)
T 3
W 4
Th 5 Exam 7 Laplace Transforms
F 6
T 10 2:50 Project presentation
Kathryn Sisson, Kevin Guthrie, Laurence Lindsey, Mashruba Tasneem
(water rocket) (filmed?)
W 11 5:40-7:00 Projects:
Justin Melvin, Jennifer Lee, Joshua Cuneo, Narendhra Seshadri
(Donnelly's Spinning Cylinder)
Saad Khan (Growing Mold)
Jeremy Corbett, Jeremy Miller (Spring-rubber band oscillator)
Th 12 2:50-5:40 Projects:
Nhan Dihn and Vishal Patel (magnetic oscillator)
Brandon Luders and Steven Lansel (Gravity fed water flow)
Christal Gordon (Modeling Neurons)
Elise Bisecker and Michael Abraham (Lorentz Water Wheel)