Curriculum.

Course title Diferencial Equations
Code M105
Status 2+0+2
Level Fundamental course
Year 2. Semester 4.
ECTS 6 ECTS credits
Lecturer Krešimir Burazin, Assistant Professor; Ivana Vuksanović, Assistant
Course objective Introduce students with the concept and geometric meaning of ordinary differential equations, and with general theorems of existence and uniqueness of solutions. Demonstrate basic types and methods for finding solution with particular emphasis on the theory of linear equations.
Prerequisites Mathematics 1. and 2.
Learning outcomes: After successfully completed course, students will be able to:

  1. identify some real world problems that can be modeled by differential equations;
  2. identify and explain the fundamental concepts, such as solution of equation, Cauchy problem, slope field and sensitivity to initial conditions;
  3. express in their own words conditions that ensure the existence (and uniqueness) of solution of Cauchy problem;
  4. solve different types of equations of the first order as well as  higher order equations that allow reduction of order;
  5. solve linear equations and systems;
Teaching activity ECTS Learning outcome Students activity Methods of evaluation Points
min max
Class attendance 2 all Class attendance Evidence list
Knowledge test (preliminary exam) 2 all Preparation for written examination Written  preliminary exam
Final exam 2 all Repetition of teaching materials Oral exam (and written exam)
Total 6
Consultations two times a week
Gained competencies Capability of modelling real-world problems with differential equations, as well as solving them.
Content (Course curriculum)
  1. Introduction. Sources of ordinary differential equations. Notion of solutions: general and particular. Cauchy problem. The geometric meaning. Problem of sensibility on change of initial conditions.
  2. Ordinary differential equations of the first order. Solution and slope field. Existence and uniqueness theorems. Some types of ordinary differential equations of the first order (exact, homogeneous, linear, Bernoulli, Lagrange, Clairaut, Riccati). Examples and applications.
  3. Ordinary differential equations of second order. Some special types. Linear differential equation of second order. Lagrange’s method of variation of constants. Linear differential equations of second order with constant coefficients. Laplace transform. Examples and applications (harmonic oscillator).
  4. Ordinary differential equations of higher order.
  5. Systems of ordinary differential equations. System of linear equations with constant coefficients. Examples and applications (ballistic problem in vacuum and air).
  6. Appendix. Partial differential equation. Concept, examples and basic methods for solving them.
Recommended reading
  1. Alić, Obične diferencijalne jednadžbe, PMF – Matematički odjel, Zagreb, 2001.
  2. Ivanšić, Fourierovi redovi. Diferencijalne jednadžbe, Department of Mathematics, University of Osijek, Osijek, 2000.
  3. W.E. Boyce, R.C. DiPrima, Elementary Differential Equations and Boundary Value Problems, 7th edition, John Wiley & Sons, 2000.
Additional reading
  1. L.E. Eljsgoljc, Differencialjnie uravnenija, Gosudarstvenoe izdateljstvo tehniko-teoretičeskoj literaturi, Moskva, 1957.
  2. G.F. Simmons, J.S. Robertson, Differential Equations with Applications and Historical Notes, $2^{nd$ Ed., McGraw-Hill, Inc., New  York, 1991.
  3. Schaum’s outline series, McGRAW-HILL, New York, 1991.
  4. S. Kurepa, Matematička analiza 2 (funkcije jedne varijable), Tehnička knjiga, Zagreb, 1990.
Instructional methods Exercises are are auditory, with usage of computer and LCD projector.
Exam formats The exam consists of a written and oral part and can be is taken after the completion of lectures and exercises. Acceptable scores on 2-4 midterm examinations, which students write during the semester, replace the written examination. Students can also make a seminar paper which can affect the final grade.
Language Croatian
Quality control and successfulness follow up Anonymous survey testing of students
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