UNIVERSITY OF NEW YORK, TIRANA

Faculty of Economics and Business Department of Mathematics and Natural Sciences

ORDINARY DIFFERENTIAL EQUATIONS and NUMERICAL METHODS (4 credits)

Spring Semester 2012

Professors:

Prof. Fejzi KOLANECI
e-mail: fkolaneci@unyt.edu.al

Office Hours: Mondays 13:00-14:00

Aim and objectives:

The course in Differential Equations is an excellent vehicle for displaying the interrelations between mathematics, engineering, physics, and biology, economic and social sciences. The interest in differential equations may be both theoretical and intensely practical (applied mathematics). The course covers topics such as classification of differential equations and historical remarks, first order differential equations, second and high order linear differential equations, series solutions of second order linear differential equations, and the Laplace transform.

Prerequisites:

Calculus II

Main textbook:

William, E. Boyce, Richard C. DiPrima, Elementary Differential Equations and Boundary Value Problems, John Wiley & Sons, Inc. New York, eighth edition, 2005.

Teaching methods:

4 hours per week in the form of lectures

Exams are closed books. Also, you use your own calculator and nothing else will be allowed. Mobile phones are strictly not tolerated in the class for any use (including computations). Active participation is meant as the effort and the interest that a student shows in the class, including homework. After each session students are expected to study all the relevant material, read all the associated exercises, identify the difficult points and

Address:
Rr.”Medar Shtylla” (ish- “Komuna e Parisit”) Tirana, Albania
Post code 1001

Telephone: +355 4 2441 330/1/2 Fax: +355 4 2441 329

Web site: http://www.unyt.edu.al

UNIVERSITY OF NEW YORK, TIRANA

pose their questions in the next session either directly to me or in the class. Cheating and plagiarism in any form will result immediately in the grade F. Students who are absent more than 20% of the total hours of the semester (i.e. 12 hours) may be required to withdraw from the course.

Assessment criteria:

Active participation 10% Homework 20% Midterm exam 30% Final exam 40%

Grading Scale:

Letter Grade

Percentage (%)

Generally Accepted Meaning

A

96-100

Outstanding work

A-

90-95

B+

87-89

Good work, distinctly above the average

B

83-86

B-

80-82

C+

77-79

Acceptable work

C

73-76

C-

70-72

D+

67-69

Work that is significantly below average

D

63-66

D-

60-62

F

0-59

Work that does not meet minimum standards for passing the course

Course outline:

 Introduction
What is an ordinary differential equation? The general solution and particular solutions. Examples. Historical remarks.

 First order differential equations
Linear equations. Separable equations. Homogenous equations. Bernoulli equations. Existence and uniqueness of solution. Exact equations. Integrating factors.

Address:
Rr.”Medar Shtylla” (ish- “Komuna e Parisit”) Tirana, Albania
Post code 1001

Telephone: +355 4 2441 330/1/2 Fax: +355 4 2441 329

Web site: http://www.unyt.edu.al

UNIVERSITY OF NEW YORK, TIRANA

 Second order linear equations
Homogeneous equations with constant coefficients. Fundamental solutions of linear homogeneous equations. The Wronskian determinant. Linear independence and the Wronskian. Complex roots of the characteristic equation Repeated roots. Nonhomogeneous equations. Method of Undetermined coefficients. Forced vibrations.

 Higher order linear equations
General theory of the n-th order linear equations, fundamental solutions. Homogeneous equations with constant coefficients.

 Series solutions to second order linear equations
Review of power series. Series solutions near an ordinary point, part 1 and part 2.

 The Laplace transform
Definition of the Laplace transform. Solution of the initial-value problems.  Numerical methods

The Euler or tangent line method. Improvements on the Euler method. The Runge- Kutta method. More on errors; Stability.

Class Conduct: You are responsible for everything that is announced, presented or discussed in class. The way to avoid any misunderstanding associated with this course is to attend class. You are expected to attend class and I do keep attendance records. Please, be courteous during class; both to me and your colleagues. I find late arrivals distracting, which cause a decline in the quality of my lecture. Importantly, it is also disruptive to your colleagues. Please, refrain from talking during class; it is disruptive to your colleagues and the lecture. I expect the best behavior from all of you. This is what education is all about. Please, consider that the language of instruction is English, so all our conversation into the class must be in this language.

If you feel that you have special learning difficulties, please, make an appointment with Dr. Enila Cenko. Dr. Enila Cenko is trained to help students with learning difficulties. She shall provide this service to our students, just as it is offered in all American universities.

Address:
Rr.”Medar Shtylla” (ish- “Komuna e Parisit”) Tirana, Albania
Post code 1001

Telephone: +355 4 2441 330/1/2 Fax: +355 4 2441 329

Web site: http://www.unyt.edu.al

Faculty: Economics.

Department: Business and Maths & N.Sciences.

Scheduele:

Instructor: Prof. Fejzi KOLANECI

Credits: 4

Prerequisite: Calculus II