Universitatea Babeş-Bolyai Cluj-Napoca
Facultatea de Matematică şi Informatică
Ciclul de studii: Licenţă
Domeniul: Calculatoare şi tehnologia informaţiei
Programul de studii: Ingineria informatiei - in limba engleza
Limba de predare: Engleză

SYLLABUS

I. General data
Code	Subject	
MME0001	Differential Equations	

Semester	Hours: C+S+L+P	Category	Status	
2	2+1+1+0	fundamental	compulsory	

II. Full status faculty members
Name and	Scientific	Didactic	Chair	Type of activity	
surname	title	title		C	S	L	
BUICA Adriana Florina	Ph.D.	Assoc.Prof.	Applied Mathematics	*			

Associated faculty members
Name and	Scientific	Institution	Type of	Type of activity	
surname	title		position	C	S	L	

III. Course objectives
Introduction to the basic problems of ordinary differential equations as well as the discussion of some mathematical models governed by differential equations.	

IV. Course contents
1.	Linear differential equations and systems (fundamental theorems and methods of finding the set of solutions).
2.	 Nonlinear differential equations (existence and uniqueness theorems for the initial value problem, direction field, Pfaff equations, first integrals, approximate solutions through numerical methods).
3.	 Continuous dynamical systems (phase portraits, singular points, stability theory, limit sets). 
4.	Discrete dynamical systems (fixed points, stability theory, limit sets).	

V. Bibliography
1.	F. Diacu, An Introduction to Differential Equations, W.H. Freeman New York, 2000.
2.	S. Lynch, Dynamical systems with applications using MAPLE, Birkhauser, 2001. 
3.	Gh. Micula, P. Pavel, Ecuatii diferentiale si integrale prin probleme si exercitii, Ed. Dacia, Cluj-Napoca,1989  
4.	I.A. Rus, Ecuatii diferentiale, ecuatii integrale si sisteme dinamice, Transilvania Press, 1996. 

VI. Thematic of didactic activities per weeks
Lecture 1. Introduction to differential equations (notions, initial conditions, boundary conditions, examples, fundamental problems). [1,3,4]
Lecture 2. Linear differential equations (existence and uniqueness theorem, fundamental theorems, the variation of constants method). [1.4]
Lecture 3. Linear differential equations with constant coefficients (the characterisitic equation method, the method of undetermined coefficients). Applications: Galileo’s pendulum. Harmonic oscillators. [1,4]
Lecture 4. Linear differential equations with variable coefficients (Euler equations, the power series method). [1,4]
Lecture 5. Linear differential systems (fundamental theorems, exponential of a matrix, reduction to a higher order differential equation). [1,4]
Lecture 6. First order one-dimensional nonlinear differential equations (Peano theorem, local and global existence and uniqueness theorem, non-uniqueness, blow-up). [1,4]
Lecture 7. Direction field. First integrals. Scalar differential equations vs. Pfaff equations vs. planar autonomous systems. [1,4]
Lecture 8. Few classes of integrable differential equations (separable, Bernoulli, Ricatti, homogeneous). Picard sequence of successive approximations. [1,4]
Lecture 9. Numerical methods for first order one-dimensional nonlinear differential equations (Euler algorithm, Runge-Kutta algorithm). [1]
Lecture 10. Higher dimensional nonlinear differential equations (existence and uniqueness theorems, successive approximations, numerical methods). [1,4]
Lecture 11. Autonomous nonlinear differential equations. Flow, orbit, phase portrait, singular points, cycles. [1,4]
Lecture 12.  Classification of singular points. Stability of singular points of linear systems and of nonlinear systems through linearization. [1,4]
Lecture 13. Discrete dynamical systems (notions, examples, fundamental problems, numerical methods vs. discrete equations). [1]
Lecture 14. Linear discrete equations. The logistic equation. [1]

Seminar 1. Linear homogeneous differential equations with constant coefficients. General solutions and properties of solutions (periodicity, oscillations, boundedness…). [1,3]
Seminar 2. Linear differential equations (the method of undetermined coefficients, the Lagrange method). [1,3]
Seminar 3. Euler equations. Power series method. Linear differential systems. [1,3]
Seminar 4. Separable equations. Direction field. Other classes of integrable differential equations. [1,3]
Seminar 5. Numerical methods. [1,3]
Seminar 6. Phase portraits. Singular points and their stability. [1,3]
Seminar 7. Discrete equations. [1,3]

Laboratory 1. Introduction to Maple. Basic notions. [2]
Laboratory 2. The use of Maple to find the general solution of linear differential equations, to solve initial and boundary value problems and to study the properties of solutions. [2]
Laboratory 3. The use of Maple to find the general solution to Euler equations and to linear systems. Power series method. Legendre equation. Hermite equation. Bessel equation. [2]
Laboratory 4. Integrable differential equations. Direction fields. [1,2]
Laboratory 5. Numerical methods. [1,2]
Laboratory 6. Phase portraits. [1,2]
Laboratory 7. Discrete equations. [1,2]	

VII. Didactic methods used
Lectures, interactive presentations, exercises, individual study, homework assignments.	

VIII. Assessment
1.	An in-class midterm exam: 10% of course grade.
2.	Lab reports, worth 15% of course grade.
3.	Final exam, worth 75% of course grade.	

IX. Additional references
1.	J. Hale, H. Kocak, Dynamics and Bifurcations, Springer Verlag New York Inc., 1991. 
2.	Jon H. Davis, Differential Equations with MAPLE: an Interactive Approach, Birkhäuser, 2001	

	Dean,							Chair head,

	Date,							Course responsible,
					Assoc.Prof. BUICA Adriana Florina, Ph.D.