Ordinary differential equations — различия между версиями

Материалы по математике, 2013-14 учебный год, НИУ ВШЭ и РЭШ
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* [http://math-hse.info/a/2013-14/mim-ode/assignment2.pdf Assignment 2] (due date: 02/24)
 
* [http://math-hse.info/a/2013-14/mim-ode/assignment2.pdf Assignment 2] (due date: 02/24)
  
=== 02/24: ODEs in arbitrary dimensions ===
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=== 02/24: ODEs in arbitrary dimension ===
* Multidimensional phase space
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* Multidimensional phase space.
* Some facts about  
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* Some facts about curves and vector-functions.
* Autonomous multidimensional ODEs
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* Autonomous multidimensional ODEs.
** Vector field
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** Vector field.
** Phase curve
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** Phase curve.
* The relation between phase curves of autonomous ODE and integral curves of corresponding non-autonomous ODE
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* The relation between phase curves of autonomous ODE and integral curves of corresponding non-autonomous ODE.
 
==== Excercises ====
 
==== 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/seminar3.pdf Problems discussed in the class]
 
* [http://math-hse.info/a/2013-14/mim-ode/assignment3.pdf Assignment 3] (due date: 03/10)
 
* [http://math-hse.info/a/2013-14/mim-ode/assignment3.pdf Assignment 3] (due date: 03/10)
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=== 03/10: 1-forms and complete differential equations ===
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* The notion of differential 1-form (covector field).
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* Direction field defined by 1-form.
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* The relation between 1-forms and differential equations.
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* Reminder: differential of a function of several variables as 1-form.
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* Complete differential equation.
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* The criterion of completeness.
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==== Excercises ====
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* [http://math-hse.info/a/2013-14/mim-ode/seminar4.pdf Problems discussed in the class]
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* [http://math-hse.info/a/2013-14/mim-ode/assignment4.pdf Assignment 4] (due date: 03/17)

Версия 23:00, 11 марта 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 in part on the following courses:
    • ODE (Math in Moscow, 2009-10) by Yury Kudryashov and Ilya Schurov
    • 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.
  • 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

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

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

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