This course expands on the ideas of calculus introduced in the calculus sequence. \u00a0The student successfully completing this course will be able to combine analytical, graphical, and numerical methods to model physical phenomena described by ordinary differential equations. \u00a0The ACU course catalog describes the course as follows:<\/p>\n
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. \u00a0Prerequisites: \u00a0MATH 186 and 187.<\/p><\/blockquote>\n
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. \u00a0 According to Edwards and Penney, in their book Differential Equations and Boundary Value Problems: Computing and Modeling, \u00a0<\/em><\/p>\n
Technical computing environments like Maple, Mathematics, and MATLAB are widely available and now used extensively by practicing engineers and scientists. \u00a0This 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. \u00a0A bonus of this more comprehensive approach is accessibility to a wider range of more realistic applications of differential equations.<\/p><\/blockquote>\n
This course is about how to predict the future. \u00a0To do so, all we have is knowledge of how things are and an understanding of the rules that govern the changes that will occur. \u00a0From 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. \u00a0Taking 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. \u00a0Our goal is to use the differential equation to predict the future value of the quantity being modeled.<\/p>\n
There are three basic types of techniques for making these predictions. \u00a0Analytic techniques involve finding formulas for the future values of the quantity. \u00a0Qualitative 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. \u00a0Numerical techniques involve doing arithmetic (or having a computer do arithmetic) calculations that yield approximations of the future values of a quantity. \u00a0We introduce and use all three approaches in the course.<\/p>\n
The course is divided into three units: \u00a0first and second order linear equations and applications, first order linear and non-linear systems, and transforms and series solutions. \u00a0The links below will take you to individual pages for each of the topics within these units. \u00a0Most pages contain lecture notes, videos, or other resources to assist you in learning the material.\u00a0 \u00a0An outline of the lecture topics with summaries is provided below.<\/span><\/p>\n
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Unit 1: \u00a0First Order Equations<\/h2>\n
Lecture 1: \u00a0Separable Equations<\/h3>\n
Lecture 2: \u00a0Slope Fields and Solution Curves<\/h3>\n
Lecture 3: \u00a0Integrating Factor Method<\/h3>\n
Lecture 4: \u00a0Autonomous Equations<\/h3>\n
Lecture 5: \u00a0Bifurcations<\/h3>\n
Lecture 6: \u00a0Changing Variables in a Differential Equation<\/h3>\n
Lecture 7: \u00a0Numerical Methods for First Order Equations<\/h3>\n
Lecture 8: \u00a0Picard’s Existence and Uniqueness Theorem<\/h3>\n
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Unit 2: \u00a0Second Order Linear Equations<\/h2>\n
Lecture 9: \u00a0Theory of Linear Differential Equations<\/h3>\n
Lecture 10: \u00a0Homogeneous Equations with Real and Repeated Roots<\/h3>\n
Lecture 11: \u00a0Introduction to Complex Variables<\/h3>\n
Lecture 12: \u00a0Homogeneous Equations with Complex Roots<\/h3>\n
Lecture 13: \u00a0Method of Annihilators<\/h3>\n
Lecture 14: \u00a0Method of Undetermined Coefficients<\/h3>\n
Lecture 15: \u00a0Spring Mass Systems and Electrical Circuits<\/h3>\n
Lecture 16: \u00a0Harmonic Oscillators<\/h3>\n
Lecture 17: \u00a0Forced Vibrations and Resonance<\/h3>\n
<\/p>\n
Unit 3: \u00a0Linear Systems of Differential Equations<\/h2>\n
Lecture 18: \u00a0Review of Linear Algebra<\/h3>\n
Lecture 19: \u00a0Matrix Representations for Linear Systems<\/h3>\n
Lecture 20: \u00a0The Eigenvalue Method for Homogeneous Systems<\/h3>\n
Lecture 21: \u00a0Repeated Eigenvalues and Generalized Eigenvectors<\/h3>\n
Lecture 22: \u00a0The Geometry of 2×2 Systems, Trace-Determinant Plane<\/h3>\n
Lecture 23: \u00a0The Fundamental Matrix<\/h3>\n
Lecture 24: \u00a0Variation of Parameters<\/h3>\n
Lecture 25: \u00a0Solutions in Terms of Matrix Exponentials<\/h3>\n
Lecture 26: \u00a0Locally Linear Systems<\/h3>\n
Lecture 27: \u00a0Qualitative Analysis of Nonlinear Systems<\/h3>\n
Lecture 28: \u00a0Nonlinear Systems and Chaos<\/h3>\n
<\/p>\n
Unit 4: \u00a0Laplace Transform<\/h2>\n
Lecture 29: \u00a0Introduction to the Laplace Transform<\/h3>\n
Lecture 30: \u00a0Transforms of Derivatives, Integrals, and Products<\/h3>\n
Lecture 31: \u00a0Inverse Laplace Transforms<\/h3>\n
Lecture 32: \u00a0Transform Solutions of IVPs<\/h3>\n
Lecture 33: \u00a0Laplace Transforms of Step Functions<\/h3>\n
Lecture 34: \u00a0Laplace Transforms of Periodic Functions<\/h3>\n
Lecture 35: \u00a0Impulses and Delta Functions<\/h3>\n
Lecture 36: \u00a0Impulsive Response Theory<\/h3>\n
Lecture 37: \u00a0Qualitative Theory of Laplace Transforms, Poles<\/h3>\n
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