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(Новая страница: «Dear '''Math in Moscow'''' students! This page will contain information related to course '''Ordinary differential equations'''. Instructor: Ilya Schurov (ilya...») |
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− | + | Dear '''Math in Moscow'''' students! | |
− | |||
− | + | This page will contain information related to course '''Ordinary differential equations'''. | |
− | + | Instructor: Ilya Schurov (ilya[http://math-info.hse.ru/wiki2013-14/index.php?title=%D0%A1%D0%BB%D1%83%D0%B6%D0%B5%D0%B1%D0%BD%D0%B0%D1%8F:%D0%97%D0%B0%D0%B3%D1%80%D1%83%D0%B7%D0%BA%D0%B0&wpDestFile=At_sign.svg Соб@ка]schurov.com). | |
− | * | + | ==References== |
+ | * Main textbook is [http://books.google.ru/books?id=JtZFAQAAIAAJ&dq=editions:tlezMQI65w0C&redir_esc=y Ordinary differential equations] by V. I. Arnold. | ||
− | * | + | * Problems were taken mostly from Problems in differential equations by A. F. Filippov. |
− | * | + | * The program and assignments are based in part on the following courses: |
+ | ** | ||
+ | ** [http://www.dyn-sys.org/wiki/MIM/2010-spring ODE] (Math in Moscow, 2009-10) by Yury Kudryashov and Ilya Schurov | ||
+ | ** | ||
+ | ** [http://math-hse.info/2012-13/ODE?uselang=en ODE] (Math in Moscow, 2013-14) by Dmitry Filimonov, Ilya Schurov and Alexandra Pushkar. | ||
+ | ** | ||
+ | ** ODE] (HSE-NES joint program, 2013-14, in Russian) by Irina Khovanskaya, Ilya Schurov, Pavel Solomatin, Andrey Petrin and Nikita Solodovnikov. | ||
− | * | + | * [http://www.mccme.ru/mathinmoscow/courses/view.php?name=Ordinary%20Differential%20Equations.htm Curriculum] (it seems that only the first 14 items will be covered in the course due to lack of time). See also the curriculum of our [http://www.dyn-sys.org/wiki/MIM/2010-spring 2009-10] course |
− | * | + | ==Lessons== |
+ | ===02/10: Introduction to ODEs=== | ||
+ | * Examples of mathematical models that lead to differential equations: Malthusian population grows, free fall, harmonic oscillator. | ||
− | * | + | * Examples of ODEs and their solutions: |
− | * | + | * Phase space, extendended phase space, direction field, integral curves. |
− | + | * Barrow's formula: the solution of an equation (autonomous equation in dimension 1). | |
− | * | ||
− | * [http://math-hse.info/ | + | ===Excercises=== |
+ | * [http://math-hse.info/a/2013-14/mim-ode/seminar1.pdf Problems discussed in the class] | ||
− | * [http://math-hse.info/ | + | * [http://math-hse.info/a/2013-14/mim-ode/assignment1.pdf Assignment 1] (due date: 02/17) |
− | * [http://math-hse.info/ | + | ===02/17: ODEs in dimension 1=== |
+ | * Example of nonuniqueness for the solution of differential equation: . | ||
+ | |||
+ | * Theorem of existence and uniqueness for | ||
+ | ** | ||
+ | ** autonomous differential equations in dimension 1 (with the proof); | ||
+ | ** | ||
+ | ** non-autonomous differential equations in dimension 1 (without the proof, it will be discussed later). | ||
+ | |||
+ | * Separation of the variables (with the proof). | ||
+ | |||
+ | ===Excercises=== | ||
+ | * [http://math-hse.info/a/2013-14/mim-ode/seminar2.pdf Problems discussed in the class] | ||
+ | |||
+ | * [http://math-hse.info/a/2013-14/mim-ode/assignment2.pdf Assignment 2] (due date: 02/24) | ||
+ | |||
+ | ===02/24: ODEs in arbitrary dimension=== | ||
+ | * Multidimensional phase space. | ||
+ | |||
+ | * Some facts about curves and vector-functions. | ||
+ | |||
+ | * Autonomous multidimensional ODEs. | ||
+ | ** | ||
+ | ** Vector field. | ||
+ | ** | ||
+ | ** Phase curve. | ||
+ | |||
+ | * The relation between phase curves of autonomous ODE and integral curves of corresponding non-autonomous ODE. | ||
+ | |||
+ | ===Excercises=== | ||
+ | * [http://math-hse.info/a/2013-14/mim-ode/seminar3.pdf Problems discussed in the class] | ||
+ | |||
+ | * [http://math-hse.info/a/2013-14/mim-ode/assignment3.pdf Assignment 3] (due date: 03/10) | ||
+ | |||
+ | ===03/10: 1-forms and complete differential equations=== | ||
+ | * The notion of differential 1-form (covector field). | ||
+ | |||
+ | * Direction field defined by 1-form. | ||
+ | |||
+ | * The relation between 1-forms and differential equations. | ||
+ | |||
+ | * Reminder: differential of a function of several variables as 1-form. | ||
+ | |||
+ | * Complete differential equation. | ||
+ | |||
+ | * The criterion of completeness. | ||
+ | |||
+ | ===Excercises=== | ||
+ | * [http://math-hse.info/a/2013-14/mim-ode/seminar4.pdf Problems discussed in the class] | ||
+ | |||
+ | * [http://math-hse.info/a/2013-14/mim-ode/assignment4.pdf Assignment 4] (due date: 03/17) | ||
+ | |||
+ | ===03/17: first integrals=== | ||
+ | * The notion of first integral | ||
+ | |||
+ | * Lie derivative along vector field | ||
+ | |||
+ | * Conservative systems with one degree of freedom. | ||
+ | |||
+ | ===Excercises=== | ||
+ | * [http://math-hse.info/a/2013-14/mim-ode/seminar5.pdf Problems discussed in the class] | ||
+ | |||
+ | * [http://math-hse.info/a/2013-14/mim-ode/assignment5.pdf Assignment 5] (due date: 03/24) | ||
+ | |||
+ | ===04/07: linear equations of first order=== | ||
+ | * Equation in variations with respect to inital condition for 1-dimensional equation. | ||
+ | |||
+ | * Linear equation of first order: homogeneous and nonhomogeneous. | ||
+ | |||
+ | * General facts about linear differential equations. | ||
+ | |||
+ | * Method of variations of parameters. | ||
+ | |||
+ | ===Exercices=== | ||
+ | * [http://math-hse.info/a/2013-14/mim-ode/assignment6.pdf Assignment 6] (due date: 04/10) | ||
+ | |||
+ | ===04/14=== | ||
+ | Linear systems on the plane with real eigenvectors. Matrix exponential. | ||
+ | |||
+ | ===04/21=== | ||
+ | Linear systems on the plane with complex eigenvectors. Calculating of matrix exponential in higher dimensions (diagonalizable and Jordan cases). | ||
+ | |||
+ | * [http://math-hse.info/a/2013-14/mim-ode/assignment7.pdf Assignment 7] (due date: 05/05) | ||
+ | |||
+ | ==Midterm== | ||
+ | * [http://math-hse.info/a/2013-14/mim-ode/midterm.pdf Midterm] |
Версия 00:31, 8 февраля 2020
Dear Math in Moscow' students!
This page will contain information related to course Ordinary differential equations.
Instructor: Ilya Schurov (ilyaСоб@каschurov.com).
Содержание
- 1 References
- 2 Lessons
- 2.1 02/10: Introduction to ODEs
- 2.2 Excercises
- 2.3 02/17: ODEs in dimension 1
- 2.4 Excercises
- 2.5 02/24: ODEs in arbitrary dimension
- 2.6 Excercises
- 2.7 03/10: 1-forms and complete differential equations
- 2.8 Excercises
- 2.9 03/17: first integrals
- 2.10 Excercises
- 2.11 04/07: linear equations of first order
- 2.12 Exercices
- 2.13 04/14
- 2.14 04/21
- 3 Midterm
References
- Main textbook is Ordinary differential equations by V. I. Arnold.
- Problems were taken mostly from Problems in differential equations by A. F. Filippov.
- The program and assignments are based in part on the following courses:
- Curriculum (it seems that only the first 14 items will be covered in the course due to lack of time). See also the curriculum of our 2009-10 course
Lessons
02/10: Introduction to ODEs
- Examples of mathematical models that lead to differential equations: Malthusian population grows, free fall, harmonic oscillator.
- Examples of ODEs and their solutions:
- Phase space, extendended phase space, direction field, integral curves.
- Barrow's formula: the solution of an equation (autonomous equation in dimension 1).
Excercises
- Assignment 1 (due date: 02/17)
02/17: ODEs in dimension 1
- Example of nonuniqueness for the solution of differential equation: .
- Theorem of existence and uniqueness for
- autonomous differential equations in dimension 1 (with the proof);
- non-autonomous differential equations in dimension 1 (without the proof, it will be discussed later).
- Separation of the variables (with the proof).
Excercises
- Assignment 2 (due date: 02/24)
02/24: ODEs in arbitrary dimension
- Multidimensional phase space.
- Some facts about curves and vector-functions.
- Autonomous multidimensional ODEs.
- Vector field.
- Phase curve.
- The relation between phase curves of autonomous ODE and integral curves of corresponding non-autonomous ODE.
Excercises
- Assignment 3 (due date: 03/10)
03/10: 1-forms and complete differential equations
- The notion of differential 1-form (covector field).
- Direction field defined by 1-form.
- The relation between 1-forms and differential equations.
- Reminder: differential of a function of several variables as 1-form.
- Complete differential equation.
- The criterion of completeness.
Excercises
- Assignment 4 (due date: 03/17)
03/17: first integrals
- The notion of first integral
- Lie derivative along vector field
- Conservative systems with one degree of freedom.
Excercises
- Assignment 5 (due date: 03/24)
04/07: linear equations of first order
- Equation in variations with respect to inital condition for 1-dimensional equation.
- Linear equation of first order: homogeneous and nonhomogeneous.
- General facts about linear differential equations.
- Method of variations of parameters.
Exercices
- Assignment 6 (due date: 04/10)
04/14
Linear systems on the plane with real eigenvectors. Matrix exponential.
04/21
Linear systems on the plane with complex eigenvectors. Calculating of matrix exponential in higher dimensions (diagonalizable and Jordan cases).
- Assignment 7 (due date: 05/05)