Ordinary differential equations — различия между версиями

Материалы по математике, 2013-14 учебный год, НИУ ВШЭ и РЭШ
Перейти к: навигация, поиск
(Lectures)
Строка 21: Строка 21:
 
* 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).
 +
 +
=== 02/17: ODE in dimension 1 ===
 +
* Example of nonuniqueness for the solution of differential equation: <math>\dot x=\sqrt[3]{x^2}</math>.
 +
* 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).
  
 
== Seminars ==
 
== Seminars ==

Версия 22:12, 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).

Lectures

02/10: Introduction to ODE

  • 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).

02/17: ODE 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).

Seminars

Assignments

assignment due date
Assignment 1 01/17/2014