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.


02/12: 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 dx/dt=f(x) (autonomous equation in dimension 1).


02/19: ODEs in dimension 1

  • Euler's approximations.
  • Example of nonuniqueness for the solution of differential equation: dx/dt=x2/3.
  • 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).
  • Separation of the variables (with the proof).


02/26: 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.


03/05: 1-forms and complete differential equations

  • Cartesian products of differential equations and separating variables.
  • From nonautonomous equation to autonomous system.
  • From higher-order ODE to a system of 1-st order ODEs.
  • 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.


03/12: First integrals

  • The notion of first integral.
  • Lie derivative along vector field.
  • Global first integrals.


03/19: Conservative systems with one degree of freedom

  • Hamilton's equation.
  • Newton's equation.
    • Examples: oscillator, linearized inverse pendulum, mathematical pendulum.
  • Kinetic and potential energy.
  • Phase portrait of Newton's equation.
  • Stable and unstable equilibria.


03/26: Midterm

04/02: Rectification theorem

  • Change of variables.
  • Rectification of the vector field near nonsingular point on the plane.
  • Normalization of the vector field on line.
  • Polar coordinates.


04/09: Linear equations in dimension 1

  • Linear equations of first order with variable coefficients.
  • Method of variation of constants.
  • Linear equations with fixed coefficients and quasipolynomial righthand-side.
    • Method of undetermined coefficients.
    • Resonances.


04/16: Planar linear systems

  • Linear transformations of linear systems of ODE with fixed coefficients.
  • Example: dx/dt=x+y, dy/dt=x-y.
  • General solution of linear system with diagonalizable matrix.
  • The relation between complex linear equation in dimension 1 and real system in dimension two with complex conjugated eigenvalues.
  • Classification of linear singular points on the plane:
    • Saddle
    • Node
    • Degenerate node
    • Dicritic node
    • Center
    • Focus
  • Stability of node and focus.


04/23: Linear systems of higher dimensions and matrix exponentials

  • Phase flow of linear system is linear map
    • Corollary: the dimension of the space of all solutions of linear system is equal to the dimension of the phase space.
  • Matrix exponential: series expansion
  • The solution of linear system as matrix exponential.
  • How to calculate matrix exponential: diagonalizable and Jordan cases.


04/30: Stability

  • Stable and unstable equilibria examples: one-dimensional equations, linear systems and conservative systems with one degree of freedom.
  • The definition of Lyapunov stability.
  • The definition of asymptotic stability.
    • Why we have to demand Lyapunov stability in the definition of asymptotic stability?


05/04: Total recall