This course expands on the ideas of calculus introduced in the calculus sequence. The student successfully completing this course will be able to combine analytical, graphical, and numerical methods to model physical phenomena described by ordinary differential equations. The ACU course catalog describes the course as follows:
MATH 361 Ordinary Differential Equations (3-0-3), fall, population models, first order differential equations, systems of first order differential equations and equilibrium points; oscillations and second order equations; Laplace transforms. Prerequisites: MATH 186 and 187.
This course will engage students in not only symbolic manipulation of various differential equations, but will heavily emphasize the role of current computing technology in obtaining qualitative as well as quantitative information. According to Edwards and Penney, in their book Differential Equations and Boundary Value Problems: Computing and Modeling,
Technical computing environments like Maple, Mathematics, and MATLAB are widely available and now used extensively by practicing engineers and scientists. This change in professional practice motivates a shift from the traditional concentration on manual symbolic methods to coverage also of qualitative and computer based methods that employ numerical computation and graphical visualization to develop greater conceptual understanding. A bonus of this more comprehensive approach is accessibility to a wider range of more realistic applications of differential equations.
This course is about how to predict the future. To do so, all we have is knowledge of how things are and an understanding of the rules that govern the changes that will occur. From calculus we know that change is measured by the derivative, and using the derivative to describe how a quantity changes is what the subject of differential equations is all about. Taking the rules that govern the evolution of a quantity and turning them into a differential equation is called modeling, and in this course we study many models. Our goal is to use the differential equation to predict the future value of the quantity being modeled.
There are three basic types of techniques for making these predictions. Analytic techniques involve finding formulas for the future values of the quantity. Qualitative techniques involve obtaining a rough sketch of the graph of the quantity as a function of time as well as a description of its long term behavior. Numerical techniques involve doing arithmetic (or having a computer do arithmetic) calculations that yield approximations of the future values of a quantity. We introduce and use all three approaches in the course.
The course is divided into three units: first and second order linear equations and applications, first order linear and non-linear systems, and transforms and series solutions. The links below will take you to individual pages for each of the topics within these units. Most pages contain lecture notes, videos, or other resources to assist you in learning the material. An outline of the lecture topics with summaries is provided below.
Unit 1: First Order Equations
Lecture 1: Separable Equations
Lecture 2: Slope Fields and Solution Curves
Lecture 3: Integrating Factor Method
Lecture 4: Autonomous Equations
Lecture 5: Bifurcations
Lecture 6: Changing Variables in a Differential Equation
Lecture 7: Numerical Methods for First Order Equations
Lecture 8: Picard’s Existence and Uniqueness Theorem
Unit 2: Second Order Linear Equations
Lecture 9: Theory of Linear Differential Equations
Lecture 10: Homogeneous Equations with Real and Repeated Roots
Lecture 11: Introduction to Complex Variables
Lecture 12: Homogeneous Equations with Complex Roots
Lecture 13: Method of Annihilators
Lecture 14: Method of Undetermined Coefficients
Lecture 15: Spring Mass Systems and Electrical Circuits
Lecture 16: Harmonic Oscillators
Lecture 17: Forced Vibrations and Resonance
Unit 3: Linear Systems of Differential Equations
Lecture 18: Review of Linear Algebra
Lecture 19: Matrix Representations for Linear Systems
Lecture 20: The Eigenvalue Method for Homogeneous Systems
Lecture 21: Repeated Eigenvalues and Generalized Eigenvectors
Lecture 22: The Geometry of 2×2 Systems, Trace-Determinant Plane
Lecture 23: The Fundamental Matrix
Lecture 24: Variation of Parameters
Lecture 25: Solutions in Terms of Matrix Exponentials
Lecture 26: Locally Linear Systems
Lecture 27: Qualitative Analysis of Nonlinear Systems
Lecture 28: Nonlinear Systems and Chaos
Unit 4: Laplace Transform
Lecture 29: Introduction to the Laplace Transform
Lecture 30: Transforms of Derivatives, Integrals, and Products
Lecture 31: Inverse Laplace Transforms
Lecture 32: Transform Solutions of IVPs
Lecture 33: Laplace Transforms of Step Functions
Lecture 34: Laplace Transforms of Periodic Functions
Lecture 35: Impulses and Delta Functions
Lecture 36: Impulsive Response Theory
Lecture 37: Qualitative Theory of Laplace Transforms, Poles