Ordinary differential equations
Материалы по математике, 2014-15 учебный год, НИУ ВШЭ
This page will contain information related to course Ordinary differential equations.
Instructor: Ilya Schurov (ilya(at)schurov.com).
- 1 References
- 2 Lessons
- 2.1 02/12: Introduction to ODEs
- 2.2 02/19: ODEs in dimension 1
- 2.3 02/26: ODEs in arbitrary dimension
- 2.4 03/05: 1-forms and complete differential equations
- 2.5 03/12: First integrals
- 2.6 03/19: Conservative systems with one degree of freedom
- 2.7 03/26: Midterm
- 2.8 04/02: Rectification theorem
- 2.9 04/09: Linear equations in dimension 1
- 2.10 04/16: Planar linear systems
- 2.11 04/23: Linear systems of higher dimensions and matrix exponentials
- 2.12 04/30: Stability
- 2.13 05/04: Total recall
- 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:
- 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).
- 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).
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.
- 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.
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.
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:
- Degenerate node
- Dicritic node
- 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.
- Problems 10 (discussed in the class)
- 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
- Problems 12 (discussed in the class)