After introducing some terminology and fundamental notions on differential equations, we explain elementary method for explicit solutions, general method for solving constant coefficient linear differential equations and some analysis on linear differential equations. This course is to be continued to Differential Equations II.
Differential equations are fundamental notions appearing in all fields of mathematics. Space of solutions have algebraic structure, existence theorems of solutions give various geometric and analytic objects of great interests. This course is an entry to these paths.
Main topic of this course is a basic theory and its applications of ordinary differential equations of one unknown variable. Ordinary differential equations describe various natural phenomena and physical laws, thus, method of solving equations and its theory are important mathematically as well as for applications. Students are expected to master method of solving differential equations, and to understand the theory to derive properties of solutions.
differential equation, initial value problem, separation of variables, complete differential equation, fundamental solution, Duhamel principle
✔ Specialist skills | Intercultural skills | Communication skills | Critical thinking skills | Practical and/or problem-solving skills |
Standard lecture course.
Course schedule | Required learning | |
---|---|---|
Class 1 | examples of differential equation, solutions of differential equation, initial value problem and boundary value problem | Details will be provided during each class session |
Class 2 | elementary method, separation of variables, homogenious case, first order linear differential equation | Details will be provided during each class session |
Class 3 | complete differential equation, non-normal form, higher order differential equation | Details will be provided during each class session |
Class 4 | linear ordinary differential equation, exponential function of matix and constant coefficient system of ordinary differential equation | Details will be provided during each class session |
Class 5 | variable coefficient system of ordinary differential equation and fundamental solution | Details will be provided during each class session |
Class 6 | non-homogeneous equation and Duhamel principle | Details will be provided during each class session |
Class 7 | solution via series expansion | Details will be provided during each class session |
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.
Introduction to differential equations, Koji Kasahara, Asakura Shoten (Japanese)
Earl A. Coddington and Norman Levinson, ``Theory of ordinary differential equations'', McGraw-Hill (1955)
Evaluation based on mid-term and final exams. Details will be provided in the class
Students are expected to have passed Calculus I / Recitation, Calculus II / Recitation, Linear Algebra I / Recitation, Linear Algebra II / Recitation.