Differential Equations

Enroll in this Course

Prerequisites: Qualifying math score and completion of Calculus BC, Linear Algebra, or equivalent

Course Format: Individually Paced

Course Length: Typically 6 months

Recommended School Credit: One full year of high school credit or one semester of college credit

Course Code: DIF

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

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.)