Ordinary differential equations — различия между версиями

Материалы по математике, 2013-14 учебный год, НИУ ВШЭ и РЭШ
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(Lectures)
(refactoring + lesson 2)
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* [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).
 
* [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).
  
== Lectures ==
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== Lessons ==
=== 02/10: Introduction to ODE ===
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=== 02/10: Introduction to ODEs ===
 
* Examples of mathematical models that lead to differential equations: Malthusian population grows, free fall, harmonic oscillator.
 
* Examples of mathematical models that lead to differential equations: Malthusian population grows, free fall, harmonic oscillator.
 
* Examples of ODEs and their solutions:
 
* Examples of ODEs and their solutions:
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* Phase space, extendended phase space, direction field, integral curves.
 
* Phase space, extendended phase space, direction field, integral curves.
 
* Barrow's formula: the solution of an equation <math>\dot x=f(x)</math> (autonomous equation in dimension 1).
 
* Barrow's formula: the solution of an equation <math>\dot x=f(x)</math> (autonomous equation in dimension 1).
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==== Excercises ====
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* [http://math-hse.info/a/2013-14/mim-ode/seminar1.pdf Problems discussed in the class]
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* [http://math-hse.info/a/2013-14/mim-ode/assignment1.pdf Assignment 1] (due date: 02/17)
  
=== 02/17: ODE in dimension 1 ===
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=== 02/17: ODEs in dimension 1 ===
 
* Example of nonuniqueness for the solution of differential equation: <math>\dot x=\sqrt[3]{x^2}</math>.
 
* Example of nonuniqueness for the solution of differential equation: <math>\dot x=\sqrt[3]{x^2}</math>.
 
* Theorem of existence and uniqueness for  
 
* Theorem of existence and uniqueness for  
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** non-autonomous differential equations in dimension 1 (without the proof, it will be discussed later).
 
** non-autonomous differential equations in dimension 1 (without the proof, it will be discussed later).
 
* Separation of the variables (with the proof).
 
* Separation of the variables (with the proof).
 
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==== Excercises ====
== Seminars ==
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* [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/seminar1.pdf Seminar 1]
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* [http://math-hse.info/a/2013-14/mim-ode/assignment2.pdf Assignment 2] (due date: 02/24)
 
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== Assignments ==
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{|class='wikitable'
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!assignment
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!due date
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|-
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|[http://math-hse.info/a/2013-14/mim-ode/assignment1.pdf Assignment 1]
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|01/17/2014
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|}
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Версия 23:35, 17 февраля 2014

Dear Math in Moscow students!

This page will contain information related to course Ordinary differential equations.

Instructor: Ilya Schurov (ilyaСоб@каschurov.com).

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 on the course Ordinary differential equations (Math in Moscow, 2010-11) by Yury Kudryashov and Ilya Schurov and the same course of 2013/14 academic year (by Dmitry Filimonov, Ilya Schurov and Alexandra Pushkar).
  • Curriculum (it seems that only the first 14 items will be covered in the course due to lack of time).

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

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