Ordinary differential equations

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Dear Math in Moscow' students!

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

Instructor: Ilya Schurov (ilya{at}schurov.com).


  • 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


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).


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).


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.


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.


03/17: first integrals

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


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.



Linear systems on the plane with real eigenvectors. Matrix exponential.


Linear systems on the plane with complex eigenvectors. Calculating of matrix exponential in higher dimensions (diagonalizable and Jordan cases).