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Differential Equations

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Eligibility: CTY-level or Advanced CTY-level math score required

Prerequisites: Successful completion of Linear Algebra or Multivariable Calculus or equivalent; Completion of both courses 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. Computer based interactives, homework and quizzes help to reinforce concepts taught in the class. Online course materials supplement the required textbook.

Each student is assigned to a CTY instructor to help them during their course. Students can contact their instructor via email with any questions or concerns at any time. Live one-on-one online review sessions can be scheduled as well to prepare for the graded assessments, which include quizzes, homework, midterm exams, and a cumulative final. Instructors use virtual classroom software allowing video, voice, text, screen sharing and whiteboard interaction.

This course does not have any synchronous class meetings, but students may schedule one-on-one virtual meetings directly with the instructor to answer questions or concerns. Videos from YouTube or other web providers may be present in the course. Video recommendations or links provided at end of videos are generated by the video host provider and are not CTY recommendations.

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.

directional field

Proctor Requirements

For enrollment starting on or after January 1, 2019, a proctor is required for this course if the student’s goal is to obtain a grade. Please review Proctor Requirements for more details.

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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Technical Requirements

This course requires a properly maintained computer with high-speed internet access and an up-to-date web browser (such as Chrome or Firefox). The student must be able to communicate with the instructor via email. Visit the Technical Requirements and Support page for more details.

Zoom online virtual classroom
This course uses an online virtual classroom which can be used for instructor-student communication if the student has any questions about the course or curriculum. The classroom works on standard computers with the Zoom desktop client and also tablets or handhelds that support the Zoom Mobile app. Students will need a computer with the Zoom desktop client installed to watch any recorded meetings. The Zoom desktop client and Zoom Mobile App are both available for free download.