Ordinary differential equations

Материалы по математике, 2013-14 учебный год, НИУ ВШЭ и РЭШ
Версия от 22:23, 8 февраля 2015; Ilya Schurov (обсуждение | вклад)

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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

03/17: first integrals

  • The notion of first integral
  • Lie derivative along vector field
  • Conservative systems with one degree of freedom.

Excercises

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

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).

Exercises

Midterm