MATH 3230.A (Fall 2025)
Ordinary Differential Equations
- MATHEMATICA resources
- While some of the homework problems will require the use
of symbolic calculations, most of them (if not all) can be done online
via publicly available Wolfram Alpha.
- Another way to avoid installing Mathematica is to log into its Cloud version: https://www.wolframcloud.com/. Sign in using your UVM email (use the option Sign in with Single Sign-on (SSO)) and then, at a prompt, UVM password and then open a new Notebook.
- If you prefer to download a copy of Mathematica, go to the UVM Software Portal,
find Mathematica, and follow your nose to install it (you will need to
enter your UVM netID and password). Between the two download options,
you can choose Manager vs Full Installer, as the former is a bit faster. After installing, the system will ask you to activate Mathematica if you don’t
already have an active license on your computer. Choose Activate Through Your Organization (SSO), enter your UVM email and follow your nose.
- There is a new feature in Mathematica 14, called
"Free-form output," available both in an installed copy of the program
and in WolframCloud (but not in Wolfram Alpha). To see how it works, at
the input begin typing something like "solve cos x + sin x = 0.5" or
"find root of x^2-1" (without quotes). A drop-down menu will appear,
one of whose options is "Convert to free-form output." Select it and
see what happens.
Lecture
Notes
Lecture
1 Introduction
Example 4 (direction field) worked out in detail
Lecture 2
Linear 1st-order differential equations
A story
of
how a simple drag force model helped resolve a controversy of an atomic waste disposal
problem (from
M. Braun, "Differential equations and their applications," 3rd Ed.,
Springer, New York, 1983)
Lecture 3
General properties of solution of 1st-order linear differential
equations
Lecture 4
Some applications of 1st-order linear differential equations
Description
of Radiocarbon
dating technique from Wikipedia
A story
about uncovering of an
art forgery
after World War II using radioactive isotope dating (from M.
Braun's textbook)
Lecture 5
Separable differential equations
Lecture
6 Existence and uniqueness of solution of 1st-order
nonlinear diferential equations
Lecture
7 Special cases when a 1st-order nonlinear
differential equation can and cannot be solved
Lecture 8 An
application of 1st-order nonlnear differential equations: The Logistic
model
Lecture 9
Euler's method
Lecture 10 Introduction
to 2nd-order differential equations: motivation; basic properties;
linear differential equations
Lecture 11 The general
solution of a homogeneous linear 2nd-order differential equation
Lecture 12 Homogeneous
linear differential equations with constant coefficients
Lecture 13 Real
repeated roots; Reduction of order of a linear differential
equation
Lecture 14 Complex
roots; Oscillatory solutions of 2nd-order diffrential equations
Lecture 15 Unforced
mechanical vibrations: Linear oscillator model revisited
Almost
all
realistic models of nature have damping in them, however small it may
be. Without damping, one would have perpetual motion. This has been
known to be impossible since the times of Isaac Newton. However, in
some very sophisticated and carefully engineered systems perpetual
motion is possible! Here is an example.
Lecture 16 General
solution of a linear nonhomogeneous differential equation
Lecture
17 Particular solution of a nonhomogeneous 2nd-order
differential equation:
Method of Undetermined Coefficients
Lecture
18 Particular solution
of a nonhomogeneous 2nd-order differential equation: Method of
Variations of
Parameters
Lecture 19 Resonance in
undamped and damped linear oscillator models
One of
the
dramatic manifestations of resonance is the destruction of Tacoma
Narrows Bridge in 1940. Here is a video of the event.
However, a connection between that event and a resonance is far from
trivial. (One obvious observation is that the resonance was not caused by a
wind that blew with periodic intensity -- such winds do not exist in
nature.) A critical review of various explanations of the Tacoma Bridge
destruction can be found in a paper by H.
Petroski, published in 1991.
Online demonstration
of the phenomenon of beats
Lecture
20 General properties of higher-order linear
differential equations
Lecture 21 Higher-order
linear differential equations with constant coefficients
Lecture 22 Systems of
first-order linear differential equations: Introduction
Lecture 23 General
properties of linear systems of first-order differential equations
Proof of Abel's Theorem for n=2
Lecture 24 Homogeneous
linear systems with constant coefficients
Lecture 25
Nonhomogeneous linear systems of differential equations: Method of
Variation of parameters
Lecture 26 Laplace
Transform: Motivation and Introduction
Lecture 27 Using Laplace Transform to solve Initial value problems