2026Year

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%

Various phenomena in engineering and information science are often modeled by differential equations. Determining their solutions and analyzing their qualitative properties are of fundamental importance not only from a mathematical perspective but also for research in engineering and information science. Ordinary differential equations in one variable are introduced at the undergraduate level in some departments, but higher-order equations are typically treated only in the linear case. However, a key feature of ordinary differential equations is that equations of order two or higher can be reduced to systems of first-order equations. In this course, after reviewing the differential equations covered at the undergraduate level, we study first-order linear systems as an application of linear algebra, and further discuss analytical methods (rather than numerical computation) for nonlinear ordinary differential equations to which quadrature methods cannot be applied.

Upon successful completion of this course, students will be able to:

(1) Solve systems of first-order linear ordinary differential equations as an application of linear algebra.
(2) Determine the stability of nonlinear differential equations.
(3) Investigate the existence of limit cycles in 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 short tests by the deadline. A report assignment will be given at the end of the semester.

The final grade for this course will be determined based on short tests and a report administered through KU-LMS, with a weighting of Short Tests : Report = 2 : 1. A short test or the report will be deemed invalid in the following cases:

(1) Failure to complete attendance registration and download the course materials by the designated deadline.
(2) When a student is judged to have copied, or allowed others to copy, answers from another student or from generative AI tools.
(3) Failure to download the report assignment file by the designated deadline.
(4) When materials other than the course materials are consulted without properly citing the source and submitting an image of the relevant page.

Evaluation of short tests and the report will be based not only on correctness but also on learning records in KU-LMS (including attendance registration, access/download history of materials, submission time, etc.). If the report is not submitted, or is deemed invalid, the final grade for the course will be recorded as F.

The correct answers for the short 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 (ft10058@g.kogakuin.jp).

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