Differential Equations

Relations:

It is not possible to register for the course 01DIFR and for the course 01ANB4 in the same semester.

It is not possible to register for the course 01DIFR if the student is concurrently registered for or has already completed the course 01ANB3 (mutually exclusive courses).

Course guarantor:

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)