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Differential Equations (NCAA Approved)

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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: Session Based. See calendar for session dates and application deadlines.

Course Length: 30 weeks

Recommended School Credit: One full year of high school credit equal to or greater than an AP class or one semester of college credit

Student Expectations: Students are strongly encouraged to work at least 1 hour a day, 5 days a week.

Course Code: DIFY

Course Description


Differential Equations is an online 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. This course covers the entire syllabus from the Johns Hopkins one-semester, in-person Differential Equations course with several additional topics. Online interactives, homework and quizzes help to reinforce concepts taught in class. Online course materials supplement the required textbook.

Online course materials, such as videos, notes, interactive webpages, and practice problems with solutions, are provided for the student. Students are expected to watch videos and review notes regularly. Each student is assigned to a CTY instructor to help them during their course.

In this course, participation in forums and synchronous online virtual sessions are required as part of the final grade. Discussion forums are located within the course where students will be required to respond to a given prompt and then comment on responses made by other students. Online sessions, led by a CTY instructor are held in Zoom, which allows for video, voice, text, screen sharing, and whiteboard interaction.

Students can contact their instructors via email with questions or concerns at any time. Live one-on-one online review sessions can be scheduled to prepare for the graded assessments, which include quizzes, homework, projects, unit exams, and a cumulative final.

The textbook must be purchased separately.

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.

This course has synchronous virtual class meetings and students may also schedule one-on-one virtual meetings directly with the instructor to answer questions or concerns.  The instructor will schedule meeting dates/times at the start of the course. Meetings will be recorded for students who are unable to attend due to scheduling conflicts.

Virtual classrooms, and student activities in the classroom, may be recorded and added to the course as an ongoing asset for all class students to review.  Students may be invited to interact in CTY community spaces that include students and instructors and potentially specially invited guests that are not enrolled in their course. 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.

Proctor Requirements

Proctor Requirements

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.