# Ordinary differential equations

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Dear Math in Moscow students!

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

## References

• 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).

### 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.

### 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:
• 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?