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Lecturer(s)
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Řezníčková Jana, Mgr. Ph.D.
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Course content
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1. Basic concepts of ordinary differential equations. 2. Separable ordinary differential equations. Separation of variables. 3. First order linear ordinary differential equations. Method of variation of a parameter. 4. Higher order linear ordinary differential equations - basic concepts and properties. Homogeneous higher order linear ordinary differential equations with constants coefficients. Characteristic equation. Fundamental solutions. 5. Nonhomogeneous higher order linear ordinary differential equations with constants coefficients. Method of variation of parameters. Method of undetermined coefficients. 6. Laplace transform - definition, basic properties. Laplace transforms of elementary functions. Inverse Laplace transform. Solving differential equations using the Laplace transform and the inverse Laplace transform. 7. Z-transform - definition, basic properties. Z-transforms of elementary functions. Inverse Z-transform. 8. Solving difference equations using the Z-transform and the inverse Z-transform. 9. Selected applications of ordinary differential and difference equations. 10. Infinite number series - basic concepts and properties. Geometric series. Tests of convergence for number series. 11. Alternating series. Absolute and conditional convergence. 12. Power series. Radius of convergence, interval of convergence of power series. 13. Taylor and Maclaurin series and their applications. 14. Trigonometric and Fourier series.
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Learning activities and teaching methods
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Lecturing, Methods for working with texts (Textbook, book), Demonstration, Projection (static, dynamic), Practice exercises
- Preparation for course credit
- 20 hours per semester
- Preparation for examination
- 40 hours per semester
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| prerequisite |
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| Knowledge |
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| Basic knowledge of linear algebra and differential and integral calculus of function of one variable is required. |
| Basic knowledge of linear algebra and differential and integral calculus of function of one variable is required. |
| learning outcomes |
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| After graduation of this course student should be able to: - define basic concepts of theory of ordinary differential equations - recognise the type of the given differential equation - choose the suitable method of solution of the given equation - solve basic types of the first order differential equations - use the method of variation of parameters and the method of undetermined coefficients - solve initial value problem using the Laplace transform for selected types of differential equations - use difference calculus in solving selected types of difference equations - apply Z-transform for solving selected types of difference equations - find the sum of infinity number series - decide on convergence (divergence) of series using the suitable test of convergence - find the radius and the interval of convergence of power series - be familiar with Maclaurin series generated by basic elementary functions - express a function as Taylor and Fourier series |
| After graduation of this course student should be able to: - define basic concepts of theory of ordinary differential equations - recognise the type of the given differential equation - choose the suitable method of solution of the given equation - solve basic types of the first order differential equations - use the method of variation of parameters and the method of undetermined coefficients - solve initial value problem using the Laplace transform for selected types of differential equations - use difference calculus in solving selected types of difference equations - apply Z-transform for solving selected types of difference equations - find the sum of infinity number series - decide on convergence (divergence) of series using the suitable test of convergence - find the radius and the interval of convergence of power series - be familiar with Maclaurin series generated by basic elementary functions - express a function as Taylor and Fourier series |
| definovat základní pojmy z teorie diferenciálních rovnic: diferenciální rovnice, řád diferenciální rovnice, řešení diferenciální rovnice, obecné řešení diferenciální rovnice, partikulární řešení diferenciální rovnice, Cauchyova úloha |
| definovat základní pojmy z teorie diferenciálních rovnic: diferenciální rovnice, řád diferenciální rovnice, řešení diferenciální rovnice, obecné řešení diferenciální rovnice, partikulární řešení diferenciální rovnice, Cauchyova úloha |
| Student defines basic concepts of theory of differential equations: a differential equation, an order of a differential equation, a solution of a differential equation, a general solution of a differential equation, a particular solution of a differential equation, an initial value problem. |
| Student defines basic concepts of theory of differential equations: a differential equation, an order of a differential equation, a solution of a differential equation, a general solution of a differential equation, a particular solution of a differential equation, an initial value problem. |
| rozpoznat obyčejnou diferenciální rovnici se separovatelnými proměnnými |
| rozpoznat obyčejnou diferenciální rovnici se separovatelnými proměnnými |
| Student recognizes an ordinary differential equation with separable variables. |
| Student recognizes an ordinary differential equation with separable variables. |
| vysvětlit, jak vypadá lineární obyčejná diferenciální rovnice prvního řádu a vyššího řádu |
| Student recognizes a first order linear ordinary differential equation. |
| vysvětlit, jak vypadá lineární obyčejná diferenciální rovnice prvního řádu a vyššího řádu |
| Student recognizes a first order linear ordinary differential equation. |
| Student defines a second order linear ordinary differential equation with constant coefficients. |
| Student defines a second order linear ordinary differential equation with constant coefficients. |
| vysvětlit význam Laplaceovy transformace při řešení diferenciálních rovnic |
| Student defines the Laplace transform and the inverse Laplace transform. |
| Student defines the Laplace transform and the inverse Laplace transform. |
| vysvětlit význam Laplaceovy transformace při řešení diferenciálních rovnic |
| Student explains the concept of an infinite number series and its sum and the convergence and the divergence of the infinite number series. |
| Student explains the concept of an infinite number series and its sum and the convergence and the divergence of the infinite number series. |
| objasnit pojmy nekonečná číselná řada a její součet, konvergence a divergence nekonečné číselné řady |
| objasnit pojmy nekonečná číselná řada a její součet, konvergence a divergence nekonečné číselné řady |
| Student defines a geometric series and its sum. |
| Student defines a geometric series and its sum. |
| Student presents some tests of the convergence of infinite number series. |
| Student presents some tests of the convergence of infinite number series. |
| Student explains the convergence of a power series. |
| Student explains the convergence of a power series. |
| Student defines the Taylor series and the Maclaurin series. |
| Student defines the Taylor series and the Maclaurin series. |
| vysvětlit pojem mocninná řada |
| vysvětlit pojem mocninná řada |
| define basic concepts of theory of differential equations: a differential equation, an order of the differential equation, a general solution and a particular solution of the differential equation, an initial value problem |
| define basic concepts of theory of differential equations: a differential equation, an order of the differential equation, a general solution and a particular solution of the differential equation, an initial value problem |
| recognize a separable differential equation |
| recognize a separable differential equation |
| explain what is a linear differential equation |
| explain what is a linear differential equation |
| explain uses of the Laplace transform in solving differential equations |
| explain uses of the Laplace transform in solving differential equations |
| define an infinity number series and its sum, convergence and divergence of the infinity number series |
| define an infinity number series and its sum, convergence and divergence of the infinity number series |
| explain the concept of a power series |
| explain the concept of a power series |
| Skills |
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| aplikovat metodu separace proměnných na řešení obyčejné diferenciální rovnice se separovatelnými proměnnými |
| aplikovat metodu separace proměnných na řešení obyčejné diferenciální rovnice se separovatelnými proměnnými |
| Student solves an ordinary differential equation with separable variables using the method of separating variables. |
| Student solves an ordinary differential equation with separable variables using the method of separating variables. |
| vyřešit lineární obyčejnou diferenciální rovnici prvního řádu metodou variace konstanty |
| vyřešit lineární obyčejnou diferenciální rovnici prvního řádu metodou variace konstanty |
| Student solves a first order linear ordinary differential equation using the method of variation of a parameter. |
| Student solves a first order linear ordinary differential equation using the method of variation of a parameter. |
| používat metodu neurčitých koeficientů při řešení lineární obyčejné diferenciální rovnice vyššího řádu s konstantními koeficienty |
| používat metodu neurčitých koeficientů při řešení lineární obyčejné diferenciální rovnice vyššího řádu s konstantními koeficienty |
| Student applies the method of undetermined coefficients in solving a higher order linear ordinary differential equation with constant coefficients. |
| Student applies the method of undetermined coefficients in solving a higher order linear ordinary differential equation with constant coefficients. |
| Student calculates Laplace domain functions for basic time domain functions using the definition integral of the Laplace transform. |
| Student calculates Laplace domain functions for basic time domain functions using the definition integral of the Laplace transform. |
| Student solves a first and a higher order linear ordinary differential equation with constant coefficients using the Laplace transform and the inverse Laplace transform. |
| Student solves a first and a higher order linear ordinary differential equation with constant coefficients using the Laplace transform and the inverse Laplace transform. |
| řešit lineární obyčejnou diferenciální rovnici prvního a vyššího řádu užitím Laplaceovy transformace |
| řešit lineární obyčejnou diferenciální rovnici prvního a vyššího řádu užitím Laplaceovy transformace |
| sečíst nekonečnou geometrickou řadu |
| sečíst nekonečnou geometrickou řadu |
| Student evaluates a sum of a geometric series. |
| Student evaluates a sum of a geometric series. |
| Student investigates the convergence of an infinite number series using a suitable test of the convergence. |
| vyšetřit konvergenci nekonečné číselné řady užitím vhodného kritéria konvergence |
| vyšetřit konvergenci nekonečné číselné řady užitím vhodného kritéria konvergence |
| Student investigates the convergence of an infinite number series using a suitable test of the convergence. |
| Student states the radius and the set of convergence of a power series. |
| Student states the radius and the set of convergence of a power series. |
| Student finds the Taylor series generated by a given function at a given point. |
| Student finds the Taylor series generated by a given function at a given point. |
| Student finds the Maclaurin series generated by a given function. |
| rozvinout danou funkci v Taylorovu řadu |
| rozvinout danou funkci v Taylorovu řadu |
| Student finds the Maclaurin series generated by a given function. |
| apply a method of separating variables in solving separable differential equations |
| apply a method of separating variables in solving separable differential equations |
| solve a first order linear differential equation using a method of variation of a parameter |
| solve a first order linear differential equation using a method of variation of a parameter |
| use a method of undetermined coefficients in solving higher order linear differential equations with constants coefficients |
| use a method of undetermined coefficients in solving higher order linear differential equations with constants coefficients |
| solve a linear differential equation using the Laplace transform |
| solve a linear differential equation using the Laplace transform |
| find a sum of a geometric series |
| find a sum of a geometric series |
| determine convergence using a suitable test of convergence |
| determine convergence using a suitable test of convergence |
| find the Taylor series for a given function |
| find the Taylor series for a given function |
| teaching methods |
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| Knowledge |
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| Lecturing |
| Practice exercises |
| Practice exercises |
| Projection (static, dynamic) |
| Projection (static, dynamic) |
| Methods for working with texts (Textbook, book) |
| Methods for working with texts (Textbook, book) |
| Demonstration |
| Demonstration |
| Lecturing |
| assessment methods |
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| Grade (Using a grade system) |
| Written examination |
| Written examination |
| Grade (Using a grade system) |
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Recommended literature
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BRONSON, Richard a Gabriel B. COSTA. Schaum's outlines of differential equations. 3rd ed. New York: McGraw-Hill, 2006. ISBN 0-07-145687-2.
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DOŠLÁ, Zuzana a Vítězslav NOVÁK. Nekonečné řady. 3. vyd.. Brno: Masarykova univerzita, 2013. ISBN 978-80-210-6416-4.
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KALAS, Josef a Miloš RÁB. Obyčejné diferenciální rovnice, 3. vydání. Brno: Masarykova univerzita, 2012. ISBN 978-80-2105-815-6.
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KELLEY, Walter G. a Allan C. PETERSON. Difference equations: an introduction with applications. 2nd ed.. San Diego: Harcourt Academic Press, 2001. ISBN 012403330x.
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NAVRÁTIL, P. Automatizace, vybrané statě. FAI,UTB ve Zlíně, 2011.
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OSTRAVSKÝ, J. Diferenciální počet funkce více proměnných. Nekonečné číselné řady. UTB Zlín, 2007.
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ŠVARC, I. Automatizace/Automatické řízení. VUT v Brně, 2005.
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VÍTEČKOVÁ, M., VÍTEČEK, A. Základy automatické regulace. VŠB TU Ostrava, 2008.
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WEIR, Maurice D., Joel HAAS, George B. THOMAS a Ross L. FINNEY. Thomas' calculus, 11th ed., media upgrade. Boston: Pearson Addison Wesley, 2008. ISBN 978-0-321-48987-6.
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