Ordinary Differential Equations

MTHE 225/3.50


Introduction to ordinary differential equations and their applications to the natural and engineering sciences.  Specific topics include first order differential equations, linear differential equations with constant coefficients, Laplace transforms, and systems of linear equations.

This course may be used by Queen’s students towards the degree requirements of programs in Arts & Science (except for MATH majors). Students from other institutions pursuing engineering or science programs should check with their home institution regarding the suitability of this course towards their degree programs.

NOTE: Some knowledge of linear algebra is assumed.

Learning Outcomes

Learning Outcomes

  • Creating differential equations from word problems/application scenarios.
  • Choosing the most appropriate method for solving a specific boundary value or initial value problem from among several different viable techniques. 
  • Generating general and particular solutions to differential equations using appropriate solving techniques.
  • Verifying that an expression or function is actually a solution to a differential equation.
  • Interpreting the results of a differential equation solution.


  • Separable 1st order DEs 
  • Linear 1st order DEs
  • Higher-order linear DEs with constant coefficients
    • homogeneous - real distinct roots, complex roots, repeated real roots.
    • non-homogeneous - overlapping and non-overlapping with the homogeneous solutions.
  • Laplace Transforms 
    • Introduction to Laplace as an integral transformation, change in domain from t to s.
    • Forward and backwards Laplace transforms of functions using table of transforms.  New functions: piecewise/step functions. 
    • Forward and backwards transformation of DEs using Laplace transforms
  • Systems of differential equations
    Background/review - row reduction, computing eigenvalues and eigenvectors of matrices
    • Transforming systems of 1st order DEs into matrix form.
    • Building vector solutions to matrix form using eigenvectors and eigenvalues.


Summer (May–July) 2023
Course Dates
Delivery Mode


Fundamentals Online Self-Test2%
Online Homework (weekly)12%
3 Tests24%
Modelling Assignments (includes peer assessment)8%
Weekly Check In4%
Proctored Final Exam50%

**Evaluation Subject to Change**

Proctored Exams

If a student is enrolled in ONLY online courses (section 700), they may choose either of the following options to write the exam:

  • Write the final exam online: you will write in onQ with Examity proctoring. A $100 online exam fee will be charged to your SOLUS account.  
  • Write the final exam in-person: you will write on Queen’s campus in Kingston. You will not be charged an extra fee to write on campus. 

If a student is enrolled in ANY in-person courses (section 001, 002, etc), you MUST write all your final exams in-person on Queen’s campus, including for an online course. You may not choose to write your exams online. 

Location and Timing of Final Exams

Once the exam schedule has been finalized the exam date will be posted on your SOLUS account. The exam dates for each Term are listed in the Academic Calendar. Student exam schedules for the Fall Term are posted via SOLUS immediately prior to the Thanksgiving holiday; for the Winter Term they are posted on the Friday before Reading Week, and for the Summer Term they are individually noted on the Arts and Science Online syllabi. Students should delay finalizing any travel plans until after the examination schedule has been posted. Exams will not be moved or deferred to accommodate employment, travel/holiday plans or flight reservations. 

Instructor Information


Textbook and Materials

ASO reserves the right to make changes to the required material list as received by the instructor before the course starts. Please refer to the Campus Bookstore website at http://www.campusbookstore.com/Textbooks/Search-Engine to obtain the most up-to-date list of required materials for this course before purchasing them.

Most learning materials (notes, practice problems, etc.) will be supplied as part of the course.

Calculator Policy
Calculators acceptable for use during quizzes, tests and examinations are intended to support the basic calculating functions required by most Arts and Science and Applied Science courses. For this purpose, the use of the Casio 991 series calculator is permitted and is the only approved calculator for this course. This calculator sells for around $25 at the Queen's Campus Bookstore, Staples and other popular suppliers of school and office supplies (http://www.queensu.ca/artsci/help/topics/calculator-policy).

Time Commitment

To complete the readings, assignments, and course activities, students can expect to spend on average, about 120 hours on the course.