Differential equations

2022/2023
Programme:
Practical Mathematics
Year:
2 year
Semester:
second
Kind:
mandatory
ECTS:
4
Language:
slovenian
Hours per week – 2. semester:
Lectures
2
Seminar
0
Tutorial
2
Lab
0
Content (Syllabus outline)

Ordinary differential equations:
Separable differential equation,
first order linear differential equation,
Euler differential equation,
Bernoulli differential equation, Ricatti differential equation, exact differential equation, existence and uniqueness of solutions.
Higher order linear differential equation:
Homogeneous equation, Wronskian, nonhomogeneuous equation, method of undetermined coefficients, method of variation of constants.
System of differential equations:
existence theorem, homogeneous and nonhomogeneous systems, phase plane, stability.
Ordinary differential equations in complex:
Linear second order differential equation, regular point of the equation, Frobenius method, Bessel's differential equation, Bessel functions, Sturm-Liouville problems, orthogonality of eigenfunctions, eigenfunction expansion (Fourier series).

Readings

M. Dobovišek, Nekaj o diferencialnih enačbah, DMFA založnišvo, Ljubljana, 2011.
E. Kreyszig: Advanced Engineering Mathematics, deveta izdaja, Wiley Publ. Inc., New York, 2006.
F. Križanič: I. Vidav. Navadne diferencialne enačbe, parcialne diferencialne enačbe, variacijski račun. DMFA Slovenije, 1991.
A. Suhadolc: Navadne diferencialne enačbe, DMFA Slovenije, 1996.
E. Zakrajšek: Analiza III, 3. izdaja, DMFA založništvo, 2002.
J. Cimprič: Rešene naloge iz Analize III. DMFA založništvo, 2001.

Objectives and competences

Students will acquire knowledge about ordinary differential equations, learn how to solve selected types of equations, with a point to linear equations. The students will be able to use the acquired knowledge at posing and resolving problems that appears in practices, such as, mechanics, environment sciences, and economics.

Intended learning outcomes

Knowledge and understanding:
Knowledge and understanding of the basic concepts of transferring non mathematical problems to a corresponding mathematical models.
Application:
Differential equations are one of the basic subjects necessary to understand mechanics and other subjects of natural, technical and social sciences. Knowledge is necessary in modelling of almost all systems.
Reflection:
Integrating theory and practical applications in solving problems.
Transferable skills:
Posing a problem, selection of a method and its application in solving the problem. Analysis of the results from the cases. Skills in using literature. Knowledge is transmitted to virtually all sciences.

Learning and teaching methods

Lectures, exercises, homeworks, consultations

Assessment

2 midterm exams instead of written exam, written exam
Oral exam
grading: 5 (fail), 6-10 (pass) (according to the Statute of UL)

Lecturer's references

Barbara Drinovec Drnovšek:
DRINOVEC-DRNOVŠEK, Barbara, FORSTNERIČ, Franc. Holomorphic curves in complex spaces. Duke mathematical journal, ISSN 0012-7094, 2007, vol. 139, no. 2, str. 203-254. [COBISS-SI-ID 14351705]
DRINOVEC-DRNOVŠEK, Barbara, FORSTNERIČ, Franc. The Poletsky-Rosay theorem on singular complex spaces. Indiana University mathematics journal, ISSN 0022-2518, 2012, vol. 61, no. 4, str. 1407-1423. [COBISS-SI-ID 16679257]
DRINOVEC-DRNOVŠEK, Barbara, FORSTNERIČ, Franc. Disc functionals and Siciak-Zaharyuta extremal functions on singular varieties. V: Proceedings of Conference on Several Complex Variables on the occasion of Professor Józef Siciak's 80th birthday : July 4-8, 2011, Kraków, Poland, (Annales Polonici Mathematici, ISSN 0066-2216, Vol. 106). Warsaw: Institute of Mathematics, Polish Academy of Sciences, 2012, str. 171-191. [COBISS-SI-ID 16436057]

Jasna Prezelj:
PREZELJ-PERMAN, Jasna. Positivity of metrics on conic neighborhoods of 1-convex submanifolds. International journal of mathematics, ISSN 0129-167X, 2016, vol. 27, no. 5, 1650047 [str. 1-24]. [COBISS-SI-ID 17704537]
PREZELJ-PERMAN, Jasna, SLAPAR, Marko. The generalized Oka-Grauert principle for 1-convex manifolds. Michigan mathematical journal, ISSN 0026-2285, 2011, vol. 60, iss. 3, str. 495-506. [COBISS-SI-ID 16134745]
PREZELJ-PERMAN, Jasna. A relative Oka-Grauert principle for holomorphic submersions over 1-convex spaces. Transactions of the American Mathematical Society, ISSN 0002-9947, 2010, vol. 362, no. 8, str. 4213-4228. COBISS-SI-ID 15641433]