Differential Equations

Garant předmětu:

Michal Beneš

Lecturer:

Michal Beneš

Tutor:

Michal Beneš, Pavel Strachota

Supervisor:

Department of Mathematics

Synopsis:

The course contains introduction in the solution of ordinary differential equations. It contains a survey of equation types solvable analytically, basics of the existence theory, solution of linear types of equations and introduction in the theory of boundary-value problems.

Requirements:

Basic course of Calculus, Linear Algebra (in the extent of the courses 01MA1, 01MAA2-4, 01LA1, 01LAA2 held at the FNSPE CTU in Prague).

Syllabus of lectures:

1. Introduction - motivation in applications

2. Basics - theory of ordinary differential equations

3. Particular types of 1st-order ODEs.

- separated and separable equations

- homogeneous equations

- equations with the rational argument of the righthand side

- linear equations

- Bernoulli equations

- Riccati equations

- Equations x=f(y') a y=f(y')

4. Existence theory for equations y'=f(x,y)

- Peano theorem

- Osgood theorem

5. Sensitivity on the righthand side and on the initial conditions

6. Linear n-th order differential equations

7. Systems of 1st order linear differential equations

8. Boundary-value problems

Syllabus of tutorials:

1. Equations with separated variables

2. Separable equations

3. Homogeneous differential equations

4. Generalized (quasi-homogeneous) differential equations

5. Equations with rational righthand-side argument s racionálním argumentem

6. Linear 1st-order differential equations

7. Bernoulli equations

8. Riccati equations

9. Differential equations x=f(y') a y=f(y')

10. Linear n-th order differential equations

with constant coefficients

11. Fundamental system for linear n-th order differential equations

12. Systems of linear 1st order differential equations with constant coefficients

Study Objective:

Knowledge:

analytical solution of selected types of equations, the basics of the existence theory, solution of linear types of equations

Skills:

Analytical solution of the known types of ordinary differential equations, mathematical analysis of the initial-value problems, solution of linear n-th order differential equations and of the system of 1st-order linear ordinary differential equations.

Study materials:

Key references:

[1] D. Schaeffer and J. Cain, Ordinary Differential Equations: Basics and Beyond, Springer-Verlag New York Inc., 2016

[2] F.Verhulst, Nonlinear Differential Equations and Dynamical Systems, Springer, Berlin 1990

[3] L.S.Pontrjagin, Obyknovennyje differencialnyje uravnenija, Nauka, Moskva 1965

[4] M.W.Hirsch and S.Smale, Differential Equations, Dynamical systems, and Linear Algebra, Academic Press, Boston, 1974

Recommended references:

[5] A. D. Polyanin and V. F. Zaitsev, Handbook of Exact Solutions for Ordinary Differential Equations, Chapman and Hall/CRC Press, Boca Raton, 2003

[6] W. Walter, Gewöhnliche Differenzialgleichungen, Springer, Berlin 1990

[7] J. Kluvánek, L. Mišík a M. Švec. Mathematics II, SVTL Bratislava 1961 (in Slovak)

[8] K. Rektorys a kol. Survey of Applied Mathematics, Prometheus, Praha 1995 (in Czech)

Note:

Time-table for winter semester 2024/2025:

Time-table is not available yet

Time-table for summer semester 2024/2025:

Time-table is not available yet

The course is a part of the following study plans:

• Aplikovaná algebra a analýza (compulsory course in the program)
• Matematické inženýrství - Matematická fyzika (PS)
• Matematické inženýrství - Matematická informatika (PS)
• Matematické inženýrství - Matematické modelování (PS)
• Mathematical Engineering - Mathematical Physics (PS)