Mathematical Analysis III

Code:

EM223

Acronym:

AM3

Instance: 2005/2006 - 1S

Active?	Yes
Web Page:	http://www.fe.up.pt/smat/
Responsible unit:	Mathematics Section
Course/CS Responsible:	Mechanical Engineering

Cycles of Study/Courses

Acronym	No. of Students	Study Plan	Curricular Years	Credits UCN	Credits ECTS	Contact hours	Total Time
LEM	227	Plano de Estudos EM Oficial a partir de 2000	2	3	6	-
		Plano para Bachareis EM a partir 2000	1	3	6	-

Teaching language

Portuguese

Objectives

The objectives are: to transmit to the students various techniques of integration of diferential equations and systems of diferential equations, including physical and geometrical interpretations.

Program

Introduction to differential equations: general classification, definition of solution and of boundary value problems. Ordinary differential equations of first order: the existence and uniqueness theorem; separable equations; homogeneous and reducible to homogeneous equations; linear equations (homogeneous and non homogeneous). Some problems modeled by first order equations: problems in mechanics, population dynamics and orthogonal trajectories. Exact equations and integrating factors. Non linear equations reducible to linear ones: the Bernoulli and the Riccatti equations. Ordinary second order differential equations reducible to first order equations: the two special cases of missing the independent variable or the dependent variable. Linear equations of order greater than one: general theory of homogeneous and non homogeneous linear nth order equations. Existence and uniqueness theorem. General solution for homogeneous linear equations with constant coefficients. Linear non homogeneous equations: the variation of parameters method. Systems of first order linear equations: introduction and its relation with an nth order linear differential equation. Some examples. Basic theory of systems of first order linear equations. Homogeneous linear equations with constant coefficients. Real or complex single eigenvalues case and repeated eigenvalues case. Fundamental matrices. The method of variation of parameters for non homogeneous systems. The Laplace transform: definition and existence conditions. Laplace transform of some basic functions using the definition. Main properties of Laplace transform: first and second translation theorems and the transform of the derivative. Inverse Laplace transform. Solution of initial value problems and of differential equations with discontinuous forcing functions, using the Laplace transform. Impulse functions and Dirac δ-function. The convolution theorem..

Mandatory literature

Luisa Madureira; Problemas de Equações Diferenciais Ordinárias e Transformadas de Laplace
Kreyszig, Erwin; Advanced Engineering Mathematics. ISBN: 0-471-59989-1

Complementary Bibliography

Krasnov, M. L.; Problemas de equações diferenciais ordinárias. ISBN: 972-9241-67-8
Boyce, William E.; Elementary differential equations and boundary value problems. ISBN: 0-471-08955-9

Teaching methods and learning activities

theoretical and practical lessons. In the theoretical classes detailed exposition of the program is presented where deduction and abstraction is fundamental.
In the theoretical-practiacal classes the students are presented with problems to solve by themselves after examples are given.

Evaluation Type

Evaluation with final exam

Assessment Components

Description	Type	Time (hours)	Weight (%)	End date
Subject Classes	Participação presencial	52,00
	Total:	-	0,00

Eligibility for exams

In each practical class the presence of each student is registered. Also some problems for individual solution are proposed, to which a positive or negative information is assigned.

Calculation formula of final grade

the classification obtained in the exam

Examinations or Special Assignments

not considered