Ordinary differential equations

Dear Math in Moscow' students!

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

Instructor: Ilya Schurov (ilya{at}schurov.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

• Problems discussed in the class

• 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

• Problems discussed in the class

• 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

• Problems discussed in the class

• 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

• Problems discussed in the class

• 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

• Problems discussed in the class

• 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)

Midterm

• Midterm