Ordinary differential equations
This page will contain information related to course Ordinary differential equations.
Instructor: Ilya Schurov (ilyaСоб@каschurov.com).
- 1 References
- 2 Lessons
- 2.1 02/10: Introduction to ODEs
- 2.2 02/17: ODEs in dimension 1
- 2.3 02/24: ODEs in arbitrary dimension
- 2.4 03/10: 1-forms and complete differential equations
- 2.5 03/17: first integrals
- 2.6 04/07: linear equations of first order
- 3 Midterm
- 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
02/10: 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.
- Assignment 6 (due date: 04/10)
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).
- Assignment 7 (due date: 05/05)