MATH  230.A  (Spring 2023)

Ordinary Differential Equations

• MATHEMATICA resources
• On that page, read the (very short) instructions, click on the link "UVM Software Portal" and follow the directions there. When prompted, you will need to login with your UVM NetID.
• After you install Mathematica on your computer, it will ask you to Activate the program and will display a window with two Activation options: (1) with an Activation Key and (ii) through your organization. Choose the latter and sign in with your UVM credentials. In this process you will again be presented with the options to sign in with your Wolfram account or via UVM; choose the latter.
• Be prepared that the installation  process takes about 15+ minutes (or longer if your connection to the Internet is slow). The entire process consists of two stages, which you will be guided through by the Installer: You first download the installation package, and then the Installer installs them on your computer.
• On the same page, notice the link to "tutorial videos" at the end. It is a very friendly resource, which will help you if you do not feel comfortable using Mathematica. For example, Tutorial 5 in that link talks about defining and plotting functions.
• You will unlikely ever need Tutorials 6 and 7.
• An alternative, but arguably a less friendly place to get started is my own introductory lab to Mathematica: On this page, you need to open Lab 1, Part 1 and work through the first half of it up to (i.e., not including) the section "Parametric plots in 2D".

•  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
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