2025Year

Second Semester

Theory of Ordinary Differential Equations

Lecture

Theory of Ordinary Differential Equations

Z1000009 Theory of Ordinary Differential Equations

HASEGAWA Kenji

2.0Credits

Mon.1Period

Shinjuku Remote

A) A high degree of specialized expertise　100%
B) The skills to use science and technology　0%
C) The ability to conduct research independently, knowledge pertaining to society and occupations, and sense of ethics required of engineers and researchers　0%
D) Creative skills in specific areas of specialization　0%

Expressing engineering and computer science models using differential equations and solving for their solutions, as well as investigating their properties, is not only of mathematical interest but also extremely important for research in engineering and computer science. While single-variable differential equations (ordinary differential equations) are covered in some undergraduate disciplines, differential equations of order two or higher are typically limited to linear equations. Ordinary differential equations have the characteristic that higher-order equations can be reduced to a system of first-order equations. In this course, after covering differential equations at the undergraduate level, this course will explain linear first-order systems (linear systems) as an application of linear algebra, as well as analytical methods for nonlinear ordinary differential equations that cannot be solved using quadrature methods (excluding numerical approaches). The objectives include:

(1) Solving systems of first-order linear differential equations as an application of linear algebra.
(2) Determining the stability of nonlinear differential equations.
(3) Examining the existence of limit cycles for nonlinear differential equations.

Understanding of calculus, linear algebra taught at the undergraduate level is necessary. It is desirable to have an understanding of exponential functions as complex functions.

Support for self-learning using ICT

1. Guidance
2. First-order differential equations (Separable form, Homogeneous form)
3. First-order linear differential equations and constant coefficient linear differential equations
4. Fundamental solutions and Wronskian
5. Linear systems and matrix exponential functions
6. Calculation of matrix exponential functions using diagonalization of matrices
7. Calculation of matrix exponential functions using Jordan canonical form
8. Calculation of matrix exponential functions when eigenvalues are complex numbers
9. Fundamental solutions of homogeneous linear systems and inhomogeneous linear systems
10. Equilibrium points and solution trajectories of autonomous systems
11. Phase portraits of two-dimensional homogeneous linear systems
12. Jacobian matrix and hyperbolic equilibrium points
13. Asymptotic stability and instability at hyperbolic equilibrium points
14. Lyapunov functions
15. Van der Pol equation and limit cycle orbit
Students will watch the slides and submit their answers to the mini-tests by the deadline. A report assignment will be given at the end of the semester.

The evaluation of grades will be based on a 3:2 ratio between the mini-tests and reports through KU-LMS. However, answers in mini-tests and reports will be considered invalid if students refer no course materials or are suspected that they allow others to copy their answer or reproduce answers from other students or AI-generated response.

The correct answers for the mini-test will be disclosed through KU-LMS. In addition to office hours, the teacher will respond to questions and opinions on Google Meet. The date and time will be discussed via email.

Differential equations for science and technology, I.Makino, K.Hasegawa, S.Takagi, Baifukan (in Japanese)

Treatises on ordinary differential equations and dynamical systems.

Tuesday, 13:00〜14:00(Hachioji:1E-313)

It is an on-demand remote class, chosen as a teaching method deemed more effective than face-to-face interaction. Unless there is a change in instructors, the class format will be maintained going forward.

Not applicable

Electrical Engineering and Electronics Program／Informatics Program