Mathematical Analysis 3

Code:

L.EC011

Acronym:

AM3

Keywords
Classification	Keyword
OFICIAL	Mathematics

Instance: 2024/2025 - 1S

Active?	Yes
Web Page:	http://moodle.up.pt
Responsible unit:	Department of Civil and Georesources Engineering
Course/CS Responsible:	Bachelor in Civil Engineering

Cycles of Study/Courses

Acronym	No. of Students	Study Plan	Curricular Years	Credits UCN	Credits ECTS	Contact hours	Total Time
L.EC	216	Syllabus	2	-	6	52	162

Teaching language

Portuguese

Objectives

OBJECTIVES:

To stimulate and motivate the student to deal with practical problems modelled by differential equations. Motivate the student to a set of analytical, numerical and qualitative techniques, fundamental to the study of the behaviour of phenomena and engineering problems modelled by differential equations.

Learning outcomes and competences

COMPETENCES AND LEARNING OUTCOMES:

Knowledge: Know and describe the fundamental concepts and methods for solving differential equations. Identify the main concepts associated to mathematical modelling using differential equations.

Understanding: Identify and interpret the different techniques to use when solving problems involving differential equations.

Application: Develop skills in solving differential equations. Knowing how to apply knowledge and the ability to understand and solve problems in new and unfamiliar situations, in broad and multidisciplinary contexts.

Analysis: Analyse, discuss and critically interpret results, highlighting the potential of methods and their limitations.

Synthesis: Formulating solutions to problems with differential equations. Combine different techniques, analytical, quantitative and numerical, in solving differential equations.

Evaluation: Criticise solutions and methodologies used. Be able to communicate their conclusions and the knowledge and reasoning behind them in a clear and unambiguous manner.

Working method

Presencial

Pre-requirements (prior knowledge) and co-requirements (common knowledge)

PREVIOUS KNOWLEDGE:
The student must have the basic knowledge of the UCs of Algebra, Mathematical Analysis 1 and 2.

Program

1. First-order differential equations
  1.1 Introduction to the study of differential equations. Basic concepts.
  1.2 Mathematical modelling and differential equations.
  1.3 Solutions, particular solution, general solution and solution set of a differential equation. Validity interval of a solution.
  1.4 Field of directions and graphs of solutions.
  1.5 Initial value problems. Existence and uniqueness of the solution of an initial value problem.
  1.6 Analytical solution.
    1.6.1 Differential equations of separable variables.
    1.6.2 Linear first-order differential equations.
    1.6.3 Change of variable in differential equations.
    1.6.4 Exact and reducible differential equations.
  1.7 Qualitative theory of autonomous differential equations.
    1.7.1 Equilibrium points.
    1.7.2 Phase line.
    1.7.3 Classification of equilibrium points in terms of stability.
  1.8 Application of differential equations to solving problems in science and engineering.

2. Differential Equations of Higher Order
   2.1 Linearly independent functions. Wronskian of two or more functions.
   2.2 Homogeneous linear differential equations.
    2.2.1 Space of solutions. Solution generator set, General solution. Dimension of the space of all solutions and order.
    2.2.2 Homogeneous linear differential equations with constant coefficients. Characteristic equation. Real and distinct roots. Complex roots. Repeated real roots. Order reduction method.
    2.2.3 Homogeneous linear differential equations with variable coefficients. Euler-Cauchy equations.
  2.3 Non-homogeneous linear differential equations.
    2.3.1 General solution.
    2.3.2 Finding a particular solution. Method of variation of parameters and method of indeterminate coefficients. Applicability of the methods in the search for a particular solution.
  2.4 Application to the study of mechanical vibrations, forced oscillations, and resonance.
  2.5 Examples of non-linear differential equations.

3. Systems of First-Order Linear Differential Equations
  3.1 Systems of first-order differential equations and applications.
  3.2 Matrices and systems of linear differential equations.
  3.3 Method of eigenvalues and eigenvectors and linear systems.
  3.4 Qualitative analysis of two-dimensional systems of linear differential equations: equilibrium points, stability and representation of the phase portrait.

4. Qualitative theory of two-dimensional systems of first-order non-linear differential equations
  4.1 Equilibrium points.
  4.2 Linearisation of non-linear systems around an equilibrium point. Classification of equilibrium points in terms of stability.
  4.3 Gradient systems and Hamiltonian systems. Properties.

5. Laplace transform and differential equations
  5.1 Definition and properties.
  5.2 Heaviside function and Dirac Delta function.
  5.3 Solving initial value problems.

 
Scientific component:80%
Technological component:20%

DEMONSTRATION OF THE SYLLABUS COHERENCE WITH THE CURRICULAR UNIT'S OBJECTIVES:

This curricular unit, essentially formative, coordinates the fundamental theoretical knowledge to study differential equations with application to several phenomena and engineering problems. The programmatic content extends the competencies for the mathematical approach to different engineering problems.

Mandatory literature

Maria do Carmo Coimbra; Equações diferenciais: uma primeira abordagem, Efeitos Gráficos Unipessoal Lda, 2022
Charles Henry Edwards; Differential Equations. ISBN: 0-13-067337-4

Complementary Bibliography

George F. Simmons, Steven G. Krantz ; trad. Helena Maria de Ávila Castro; Equações diferenciais. ISBN: 978-85-86804-64-9
Stewart, James 1908-1997; Cálculo. ISBN: 85-211-0484-0
Adkins, William, Davidson, Mark G. ; Ordinary Differential Equations, Springer-Verlag New York, 2012. ISBN: 978-1-4614-3618-8 (Access to this content is enabled by Universidade do Porto)
Paul Blanchard; Differential equations. ISBN: 0-495-01265-3

Teaching methods and learning activities

All topics of the course unit are exposed in theoretical and practical classes. The theoretical exposition classes consist of oral presentations where deduction and abstraction are considered fundamental. In the lecture classes emphasis is given to the exposition of concepts, principles and theories, making frequent use of physical and geometric examples.  In the theoretical-practical classes, the problems proposed in the exercise sheets are discussed and the students are encouraged to solve them individually or in groups. The classes are complemented with a Moodle page where, besides all the pedagogical support material, self-assessment tests are available on-line to allow the evaluation of the teaching/learning process. The use of software (Octave/Matlab) is encouraged and numerical simulation is presented whenever appropriate.

DEMONSTRATION OF THE COHERENCE BETWEEN THE TEACHING METHODOLOGIES AND THE LEARNING OUTCOMES:

Students are motivated to apply their knowledge and ability to understand and solve problems described by differential equations in new situations, in large and multidisciplinary contexts.

Software

Octave
Matlab
Jupyter Notebook

keywords

Physical sciences > Mathematics > Mathematical analysis > Differential equations

Evaluation Type

Distributed evaluation with final exam

Assessment Components

Designation	Weight (%)
Teste	20,00
Exame	70,00
Trabalho prático ou de projeto	10,00
Total:	100,00

Amount of time allocated to each course unit

Designation	Time (hours)
Estudo autónomo	110,00
Frequência das aulas	52,00
Total:	162,00

Eligibility for exams

According to the regulations and directives of the L.EC Director.

Exemption from attendance: ‘In the current academic year (2024/2025), students who have attended the UC in a previous academic year are exempt from the criterion of obtaining attendance in the current academic year, for all purposes. From the next academic year forward, students who have obtained a final mark of 6 or more in the UC in the immediately preceding academic year will be exempt from the criterion of obtaining attendance in that academic year, for all intents and purposes.’

Calculation formula of final grade

The formula of calculation of the final classification for grades higher or equal to 7.5 in the Final Examination is:

CF = maximum { EF; 0.7xEF + 0.2xTS + 0.1xQZ}

where,

EX - final exam classification, onsite Test 
TS - grade of the Summative, onsite Test
QZ - Average of the marks in 3 online activities (quizzes)

For exam grades lower than 7.5, the final grade is the exam grade EF.


Distributed assessment obtained in previous years is not valid.

 

Special assessment (TE, DA, ...)

Final Exam.

Classification improvement

Final Exam.