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Differential Equations

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Prerequisites: Qualifying math score and completion of Calculus BC. Completion of Linear Algebra or Multivariable Calculus preferred.

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 equal to or greater than an AP class

Course Code: DIF

Course Description

Description

This course is equivalent to the final course in a typical college-level calculus sequence. It will cover all topics covered in JHU's Ordinary Differential Equations course. 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. Students are expected to read from the text regularly and are encouraged to attempt a few practice problems each day. Questions and hints can be obtained by direct weekly communication with instructors who provide continual feedback on their work. Students are strongly encouraged to work on the course at least 1 hour a day, 5 days a week, and email their instructors at least once per week.

Students explore autonomous differential equations with dfield.
Students explore autonomous differential equations with dfield. http://math.rice.edu/~dfield/dfpp.html

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

Differential Equations textbook cover

A graphing calculator is required, such as:

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

*Recommended

List of Topics

Upon successful completion of the course, students will be able to demonstrate mastery over the following 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

This course requires high-speed Internet access (such as Cable or DSL) for online lesson videos. Your browser will need to allow javascript, login cookies, and popup windows from ctyjhu.org, bluejay.cty.jhu.edu, and any other course web sites.

Students will also need:

  • Regular access to email for communication with instructor
  • Adobe Reader for PDFs

This course uses an online classroom for individual or group discussions with the instructor. The classroom works on standard computers with the Adobe Flash plugin, and also tablets or handhelds that support the Adobe Connect Mobile app.