Mathematical Analysis 3

Code:

L.EC011

Acronym:

AM3

Keywords
Classification	Keyword
OFICIAL	Mathematics

Instance: 2025/2026 - 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	184	Syllabus	2	-	6	52	162

Teaching language

Portuguese

Objectives

OBJECTIVES:

Acquire theoretical and practical knowledge of analytical, numerical and qualitative techniques for solving engineering problems modeled by ordinary differential equations: determine the solutions of 1st order differential equations using analytical techniques (equations of separable variables, exact equations, linear equations); apply the theorem of existence and uniqueness of the solution of an initial value problem; determine numerical solutions (Euler methods); use scientific computing, MATLAB or Python, to determine numerical solutions; determine the solutions of 2nd order linear differential equations and systems of 1st order linear differential equations; discuss the behavior of solutions; apply qualitative theory to study the behavior of engineering problems modeled by systems of linear differential equations; determine and classify equilibrium points; draw and analyze trajectories; use symbolic calculus to solve problems.

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 Computation, Algebra, Numerical Analysis and 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 Laplace transform and 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.

 
Scientific component:80%
Technological component:20%

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

The course syllabus includes the mathematical concepts and methods necessary for students to achieve the objectives of this course. These contents promote an understanding of mathematics as an essential language and tool for engineering practice and are organized into topics that adequately cover the established objectives. Students are encouraged to apply their knowledge and ability to understand and solve problems described by differential equations in new situations, in broad and multidisciplinary contexts, using analytical, numerical and qualitative techniques, making use of computational, numerical and symbolic tools, allowing the development of critical thinking skills and problem-solving strategies, in class and in autonomous work supported and accompanied based on activities made available on Moodle.

Mandatory literature

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

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

In lectures, the subject matter is presented using oral presentations, short videos, and activities specially designed for the topic in question. In oral presentations, mathematical reasoning, deduction, and abstraction are considered fundamental and are valued. Emphasis is placed on the presentation of concepts, principles and theories, making frequent use of physical and geometric examples. In theoretical-practical classes, discussion of the problems proposed in the worksheets is promoted, and students are encouraged to solve these problems individually or in groups. The classes are complemented by a Moodle page where, in addition to all the pedagogical support material, online self-assessment tests are made available to assess teaching/learning. The use of software (MATLAB or Python) is encouraged, and numerical simulation is presented whenever appropriate. In addition, explanatory videos and problem-solving demonstrations are used with the support of Artificial Intelligence tools, promoting more interactive learning and keeping up to date with emerging technologies. Theoretical and theoretical-practical classes offer activities that encourage students to study independently. They are also motivated to apply their knowledge and ability to understand and solve problems described by differential equations in new situations, in broad and multidisciplinary contexts.

Software

Octave
Matlab
Jupyter Notebook
Python

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:

Approval of the curricular unit implies compliance with the attendance condition, with it being considered that a student meets this condition if, having been regularly enrolled, they do not exceed the maximum number of absences corresponding to 25% of the scheduled in-person classes for each type. In addition to the cases provided for by statute in the rules in force at FEUP, students who obtained a final grade in the UC equal to or greater than 6 points in the immediately preceding academic year are exempt from the attendance requirement for the curricular unit.

Calculation formula of final grade

The formula of calculation of the final classification for grades higher or equal to 6 in the Final Examination is:

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

where,

EF - 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 6, the final grade is the exam grade EF.


The classification of the distributed assessment obtained in previous years is not valid.

 

Special assessment (TE, DA, ...)

Final Exam.

Classification improvement

Final Exam.

Observations

The assessments for this course are dane on computers in the Moodle environment. The use of calculators is not permitted.

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

During any assessment period, possession of any electronic devices (e.g., mobile phones, tablets, headphones, smartwatches, etc.) is strictly prohibited.
It is the responsibility of the student to anticipate this situation before the start of the assessment period.