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Differential Equations

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Prerequisites: Qualifying math score and completion of AP 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 equal to or greater than an AP class or one semester of college credit

Course Code: DIF

Course Description


Differential Equations is an online and individually-paced course equivalent to the final course in a typical college-level calculus sequence. This course is a broad introduction to Ordinary Differential Equations, and covers all topics in the corresponding course at the Johns Hopkins Krieger School of Arts and Sciences.

The course is based on a required textbook and around student-instructor interaction where CTYOnline instructors provide individual feedback on tests, answer questions about the course material, and hold live review sessions in an online virtual classroom as needed.  Computer based interactives, homeworks and quizzes help to reinforce concepts taught in the class.

Topics include:

  • First order differential equations
  • Linear differential equations
  • Series solutions of second order linear equations
  • The Laplace transform
  • First order systems (both linear and nonlinear)

For a detailed list of topics, click the "List of Topics" tab.

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Materials Needed

A textbook purchase is required for this course:

Elementary Differential Equations by Boyce and DiPrima, 10th Ed.

Elementary Differential Equations textbook cover
ISBN-10: 0470458321
ISBN-13: 978-0470458327

For those students who plan to continue their studies of differential equations beyond this course, we recommend the text Elementary Differential Equations and Boundary Value Problems, which contains the entire content of the book listed above, as well as two extra chapters not covered in this course.

Only one of the textbooks listed on this page is required for this course.

Elementary Differential Equations and Boundary Value Problems textbook cover

ISBN-10: 0470458313
ISBN-13: 978-0470458310

List of Topics

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

First Order Differential Equations

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

Linear Differential Equations

  • Review of Matrices
  • The Wronskian
  • nth Order Linear Equations
  • Homogeneous Equations with Constant Coefficients
  • Complex Roots of the Characteristic Equation
  • Repeated Roots of the Characteristic Equation
  • Higher Order Homogeneous Equations
  • Undetermined Coefficients
  • Variation of Parameters

Advanced Solving Techniques

  • Review of Power Series
  • Solutions Near an Ordinary Point
  • Euler Equations and Regular Singular Points
  • Definition of Laplace Transform
  • Solving IVPs with the Laplace Transform
  • Step Functions
  • Discontinuous Forcing Functions
  • Impulse Functions
  • The Convolution Integral

Linear Systems of Differential Equations

  • Introduction to First Order Systems
  • Eigenvalues and Eigenvectors
  • First Order Linear Systems
  • Homogeneous Linear Systems with Constant Coefficients
  • Complex Eigenvalues
  • Repeated Eigenvalues
  • The Phase Plane
  • Fundamental Matrices
  • Nonhomogeneous Linear Systems

Nonlinear Systems

  • 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 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 CTY course web sites.

You may also need the Java Runtime Environment.