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

02/12: 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).

Assignments

• Assignment 1 (due date: 02/19)

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).

Excercises

• Problems 2 (discussed in the class)

• Assignment 2 (due date: 02/26)

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.

Excercises

• Problems 3 (discussed in the class)

• Assignment 3 (due date: 03/05)

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.

Exercises

• Problems 4 (discussed in the class)

• Assignment 4 (due date: 03/12)

03/12: First integrals

• The notion of first integral.

• Lie derivative along vector field.

• Global first integrals.

Exercises

• Problems 5 (discussed in the class)

• Assignment 5 (due date: 03/19)

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.

Exercises

• Problems 6 (discussed in the class)

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.

Exercises

• Problems 7 (discussed in the class)

• Assignment 6 (due date 04/09)

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.

Exercises

• Problems 8 (discussed in the class)

• Assignment 7 (due date 04/16)

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.

Exercises

• Problems 9 (discussed in the class)

• Assignment 8 (due date 04/23)

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.

Exercises

• Problems 10 (discussed in the class)

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?

Exercises

• Problems 11 (discussed in the class)

• Assignment 9 (due date 05/07).

Exercises

• Problems 12 (discussed in the class)