Math 2080 (Honors), Fall 2017
Math 2080 (Honors), Fall 2017
Class: Introduction to Ordinary Differential Equations.
Instructor: Dr. Macauley
Resources
Lectures and Worksheets
When I
taught Math 2080 online, I made 52 lectures and posted them
on YouTube. You can view the full 32-hour, 33-minute
playlist here, or find the individual lectures below. I have
also posted the slides, but they have limited use on their own,
because I write all over them in the YouTube lectures, and this
handwriting does not appear in the pdf version. A few minor errors
have been found in the videos, which are mentioned in the video
description and/or comments.
I am linking to the videos here to serve as a supplemental resource,
with the understanding that they are not intended as a
substitute for coming to class. Rather, they should be used if you
want to re-watch a particular example or concept, or if you have to
miss class due to illness. To avoid blurriness, these are best viewed
by changing the settings to 720p (High Definition) rather than the
default of 240p. This can be easily done by clicking the "wheel" on
the lower right corner; right next to the "cc" button.
Section 1: Introduction to Ordinary Differential Equations. (3
lectures: 1 hr 23 min). Modeling physical situations that exhibit
exponential growth and exponential decay. Plotting slope fields using
the isocline method. Sketching slope fields of autonomous differential
equations. Approximating solutions using Euler's method.
Lecture notes. 9 pages, last updated 1/21/11. Brannan/Boyce: Sections 1.1--1.3, 2.3, 8.1, supplemental material.
Section 2: First Order Differential Equations. (8 lectures: 4
hrs 49 min). Solving 1st order ODEs using separation of variables, the
integrating factor method, and variation of parameters. Structure of
solutions to 1st order linear ODEs, and connections to parametrized
lines. Models of motion with air resistance. Mixing problems. The
logistic equation as a population model.
Lecture notes. 21 pages, last updated 2/17/11. Brannan/Boyce: Sections 2.1--2.6.
Section 3: Second Order Differential Equations. (9 lectures: 6
hrs 24 min). Models that use 2nd order ODEs. Solving homogeneous
linear 2nd order ODEs. Solving inhomogeneous ODEs using the method of
undertermined coefficients. Simple harmonic motion. Harmonic motion
with damping and with forcing terms. The variation of parameters
method for 2nd order ODEs. Solving 2nd order non-constant coefficient
ODEs. Cauchy-Euler equations. The power series method, and the theorem
of Frobenius.
Lecture notes. 29 pages, last updated 2/17/11. Brannan/Boyce: Sections 4.1--4.7, 9.1--9.6.
Section 4: Systems of Differential Equations. (9 lectures: 5
hrs 29 min) Intro to linear algebra: Adding and multiplying
matrices. Writing systems of linear equations with matrices, inverses
and determinants of 2x2 matrices, eigenvalues and eigenvectors of 2x2
matrices. Using linear algebra to solve systems of two 1st order
linear ODEs x'=Ax; 3 cases (i) real distinct eigenvalues, (ii)
repeated eigenvalues, (iii) complex eigenvalues. The SIR model in
epidemiology.
Lecture
notes. 26 pages, last updated 10/20/10. Brannan/Boyce:
Sections 3.1--3.6, 4.7, A.1.
Section 5: Laplace Transforms. (6 lectures: 3 hrs 53
min). Definition and properties of the Laplace transform. Using
Laplace transforms to solve ODEs. Using the Heavyside function to
express, and take the Laplace transform of, piecewise continuous
functions. Solving ODEs with discontinuous forcing terms. Taking the
Laplace transform of periodic functions. Impulse functions and delta
functions. Convolution.
Lecture notes. 21 pages, last updated 6/24/13. Brannan/Boyce: Sections 5.1--5.8.
Section 6: Fourier Series & Boundary Value Problems. (6
lectures: 3 hrs 28 min). Introduction to Fourier series -- derivation
and computation. Even and odd functions, and Fourier cosine and sine
series. Complex version of Fourier series. Parseval's
identity. Applications to summing series and to solving ODEs. Boundary
values problems.
Lecture notes. 13 pages, last updated 12/9/11. Brannan/Boyce: Sections 10.1--10.3.
Section 7: Partial Differential Equations. (8 lectures: 4 hrs 3
min). The (1-dimensional) heat, transport, and wave
equations. Analysis of different boundary conditions. Introduction to
PDEs in higher dimensions. Harmonic functions, Laplace's equation, and
steady-state solutions to the heat equation. Solving Laplace's
equation, the heat equation, and the wave equation in two
dimensions.
Lecture notes. 23 pages, last updated 7/29/10. Brannan/Boyce: Sections 11.1--11.4, 11.6, 11.A, 11.B
Section 8: Systems of Nonlinear Differential Equations The SIR
model in epidemiology. Models for competing species and predatory-prey
equations in population dymamics. Linearizing a nonlinear system at
steady-state solutions.
Lecture notes. 14 pages, last updated 12/3/15. Brannan/Boyce: Sections 7.2--4, 7.P.1