Differential Equations

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Course Description

Description

This course is equivalent to the final course in a typical college-level calculus sequence. Applications include Newtonian physics, electrical circuits, and population dynamics.

Topics include:

• first order differential equations
• second order linear equations
• series solutions of second order linear equations
• Laplace transforms,systems of first order linear equations
• an introduction to numerical methods
• stability questions for nonlinear differential equations

Assignments are based on a textbook that is purchased separately by the student.

Materials Needed

A textbook purchase is required for this course:

Elementary Differential Equations by Boyce and DiPrima, 9th Ed.

• ISBN-10: 0470383348
• ISBN-13: 978-0-470-38334-6

A graphing calculator is required, such as:

• TI-83 PLUS
• TI-84 PLUS
• TI-85
• TI-86
• TI-89*

*Recommended

List of Topics

First Order ODEs

• Separable Equations
• First Order Linear ODEs
• Exact Equations
• Euler's Method
• Autonomous Equations
• Existence & Uniqueness

Higher Order ODEs

• Review of Matrices
• Linear Independence, Eigenvalues, and Eigenvectors
• The Wronskian
• nth Order Linear Equations
• Homogeneous Equations with Constant Coefficients
• Complex Roots
• Repeated Roots
• Higher Order Homogeneous Equations
• Undetermined Coefficients
• Variation of Parameters
• Higher Order Undetermined Coefficients
• Higher Order Variation of Parameters
• Linear Differential Operators

Fancy Techniques for Solving ODEs

• Review of Power Series
• Solutions Near an Ordinary Point
• Euler Equations, and Regular Singular Points
• Definition of Laplace Transform
• Solving Initial Value Problems
• Step Functions
• Discontinuous Forcing Functions
• Impulse Functions
• The Convolution Integral
• Legendre Polynomials

Systems of ODEs

• Introduction to Systems
• Basic Theory of Systems
• Homogeneous Linear Systems with Constant Coefficients
• Complex Eigenvalues
• Repeated Roots
• Electrical Circuits
• The Phase Plane
• Fundamental Matrices
• Nonhomogeneous Linear Systems
• Matrix Exponentials

Nonlinear Differential Equations

• Autonomous Systems and Stability
• Locally Linear Systems
• Competing Species
• Predator-Prey Relationships
• Periodic Solutions and Limit Cycles

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System Requirements

CTYOnline courses require a properly-maintained computer with Internet access and a recent-version web browser (such as Firefox, Safari, or Internet Explorer) with the Adobe Flash plugin. Students are expected to be familiar with standard computer operations (e.g. login, cut & paste, email attachments, etc).

This course uses an online mathematical whiteboard for individual or group discussions with the instructor. The whiteboard web site requires cookies, popup windows, and the Java Runtime Environment. (Note: iOS & Android devices cannot run Java applets.)