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Lecturer(s)
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Daněk Josef, Doc. Ing. Ph.D.
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Tomiczek Petr, RNDr. CSc.
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Šebková Milena, RNDr.
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Kopincová Hana, Ing. Ph.D.
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Course content
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Week 1: ODEs of the 1st order, nonlinear, linear. Physical motivation (RC circuit). General and particular solutions, singular solutions. Formulation of the initial value problem. Week 2: Methods of solving ODEs of the 1st order: direct integration, separation of variables, variation of parameters. First order linear ODEs. Physical motivation (RL circuit). Week 3: Linear ODEs of higher orders - homogeneous, nonhomogeneous, with constant coefficients. Physical motivation (RLC circuit) Method of characteristic equation. Week 4: Variation of parameters. Estimate of particular integral. Week 5: Systems of ODEs of the 1st order. Physical motivation (inductively connected RL circuits). Vector functions of one real variable (limit, continuity, derivative, parametric curves). Week 6: Systems of ODEs of the 1st order. Fundamental matrix. Variation of parameters. Week 7: Boundary value problems. Eigenvalue problems. Week 8: Direct Laplace transform with a real parameter and its proerties. Week 9: Inverse Laplace transform. Week 10: Application of Laplace transform to initial value problems for ODEs. Fourier transform. Week 11: Taylor series. Week 12: Fourier series. Week 13: Recapitulation.
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Learning activities and teaching methods
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Interactive lecture, Lecture with practical applications, Practicum
- Preparation for an examination (30-60)
- 32 hours per semester
- Preparation for formative assessments (2-20)
- 20 hours per semester
- Contact hours
- 52 hours per semester
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| prerequisite |
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| Knowledge |
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| state the Taylor's theorem |
| describe the derivative and the integral of a real-valued function of one real variable |
| describe a sequence and a series of real numbers |
| describe a continuous function and the inverse function |
| use actively vectors and matrices |
| Skills |
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| calculate derivatives and integrals of basic functions of one real variable |
| draw the graphs of inverse functions; algebraic functions; goniometric functions; exponential and hyperbolic functions |
| establish convergence and divergence of a sequence, a series, and an integral |
| calculate the determinant of a matrix |
| find eigenvalues and eigenvectors of a matrix |
| Competences |
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| N/A |
| N/A |
| learning outcomes |
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| Knowledge |
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| formulate the basic initial and boundary value problems for ODEs |
| define Laplace transform and describe its properties |
| define Fourier transform |
| define Taylor and Fourier series of a function |
| describe a vektor-valued function of one real variable and a parametric curve |
| Skills |
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| solve ODEs of the first order |
| solve linear ODEs of higher orders with constant coefficients |
| solve systems of linear ODEs of the first order with constant coefficients |
| apply the Laplace transform to solve the initial value problems. |
| apply ordinary differential equations and their solutions to real problems |
| solve the boundary value problems |
| find the Taylor and Fourier expansion of basic functions |
| Competences |
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| N/A |
| N/A |
| teaching methods |
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| Knowledge |
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| Interactive lecture |
| Practicum |
| Skills |
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| Practicum |
| Task-based study method |
| Competences |
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| Lecture |
| Practicum |
| assessment methods |
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| Knowledge |
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| Combined exam |
| Test |
| Skills demonstration during practicum |
| Skills |
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| Oral exam |
| Written exam |
| Skills demonstration during practicum |
| Competences |
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| Oral exam |
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Recommended literature
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Coddington, Earl; Carlson, Robert. Linear ordinary differential equations. Philadelphia, 1997. ISBN 0-89871-388-9.
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Kufner, Alois. Obyčejné diferenciální rovnice. 1. vyd. Plzeň : Západočeská univerzita, 1993. ISBN 80-7082-106-X.
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Míka, Stanislav; Kufner, Alois. Okrajové úlohy pro obyčejné diferenciální rovnice. 2. upr. vyd. Praha : SNTL - Nakladatelství technické literatury, 1983.
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Nagy, Jozef. Soustavy obyčejných diferenciálních rovnic : Vysokošk. příručka pro vys. školy techn. směru. 2., nezm. vyd. Praha : SNTL, 1983.
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