2021 Ordinary Differential Equations

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Academic unit or major
Undergraduate major in Mechanical Engineering
Yamazaki Takahisa  Yoshida Kazuhiro 
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Course description and aims

This course focuses on the ordinary differential equations used in the analysis of linear systems and nonlinear systems. Topics include 1st order ordinary differential equations, n-th order ordinary differential equations, Fourier series, etc. and also include the functions of elementary solutions of ordinary differential equation. By combining lectures and exercises, the course enables students to understand and acquire the fundamentals of mathematical tools widely applicable to linear systems and nonlinear systems in engineering.
This course covers fundamentals of ordinary differential equations as a mathematical knowledge required to solve problems and develop mechanical engineering. By using a mathematical approach such as the differential operation method, students will experience the satisfaction of solving practical problems by using their mathematical knowledge acquired through this course.

Student learning outcomes

By the end of this course, students will be able to:
1) Explain linear systems and nonlinear systems in ordinary differential equations.
2) Explain the properties of elementary solutions using Wronski determinant.
3) Explain solving method of n-th order ordinary differential equations.
4) Explain the fundamentals of the functions of f general solutions.
5) Apply differential operator to solve problems.


Ordinary differential equations, elementary solutions, differential operator, Fourier series, nonlinear ordinary differential equations, perturbation method.

Competencies that will be developed

Specialist skills Intercultural skills Communication skills Critical thinking skills Practical and/or problem-solving skills
For applying to mechanical systems, the knowledgs of ordinary differential equations become the foundation of the ability of explanation using mathematics.

Class flow

At the beginning of each class, solutions to exercise problems that were assigned during the previous class are reviewed. Towards the end of class, students are given exercise problems related to the lecture given that day to solve. To prepare for class, students should read the course schedule section and check what topics will be covered. Required learning should be completed outside of the classroom for preparation and review purpose.

Course schedule/Required learning

  Course schedule Required learning
Class 1 Introduction of ordinery differntial equations. 1st order ordinery differntial equations Understand elementary solutions, particular solutions, and general solutions.
Class 2 n-th order ordinery differential equations Understand general solutions and Wronski determinant.
Class 3 n-th order ordinery differential equations Understand differential operator and variation of parameters.
Class 4 Similutanious ordinary differential equations Understand eigenvalue.
Class 5 Solving by series Understand solving by series.
Class 6 Nonlinear ordinary differential equations, orthogonal functions Understand perturbation method and Bessel function.
Class 7 Fourier series Understand Fourier series

Out-of-Class Study Time (Preparation and Review)

To enhance effective learning, students are encouraged to spend approximately 100 minutes preparing for class and another 100 minutes reviewing class content afterwards (including assignments) for each class.
They should do so by referring to textbooks and other course material.


Kentaro Yano and Shigeru Ishihara: Basic Analysis, Shokabo Co., Ltd, .ISBN: 978-4-7853-1079-0. (Japanese)

Reference books, course materials, etc.

Osamu Takenouchi: Differential Equations and Their Application, Saiensu-sha Co., Ltd. Publishers, ISBN: 4-7819-1060-2. (Japanese)

Assessment criteria and methods

Students' knowledge of 1st order ordinary differential equations, n-th order ordinary differential equations, and Fourier series, etc., and their ability to apply them to problems will be assessed.
レポート 40%, exercise problems 60%.

Related courses

  • MEC.B213 : Partial Differential Equations

Prerequisites (i.e., required knowledge, skills, courses, etc.)

Students must have knowledge of calculus.


Term examination to solve problems related to lectures is not scheduled because of corona virus.

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