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Differential Equations and Modelling I

Code: 100100
Credits: 9
2026/2027
Degree programme Type Course
Mathematics OB 3

Contact lecturer

Name :
Jordi Villadelprat Yague
Email :
jordi.villadelprat@uab.cat

Teaching staff

Joan Torregrosa Arus

Group languages

You can consult this information at the end of the document.

Prerequisites

Linear Algebra

Fundaments of mathematics

Calculus in 1 and several real variables.

Objectives

The theory of Differential Equations (DEs) is distinguished both by the richness of its ideas and methods and by its applicability. Thus, the subject Differential Equations and Modelling I has a theoretical component (which will be developed in theory and problem-solving classes) and a highly applied component, which will be introduced in theory classes and practised in both problem-solving classes and practical sessions held in the computer laboratory. On one hand, emphasis will be placed on the presentation of theory and the proof of results; on the other hand, students will learn to model real-life situations that allow them to predict the behaviours under study.

From a educational standpoint, we believe this subject is valuable for showing students that certain theoretical results they are already familiar with from other courses (such as topological properties of normed spaces or the Jordan canonical form theorem) are applied when laying the foundations of the theory of differential equations, ultimately enabling them to answer questions motivated by applied problems governed by deterministic models.


Learning outcomes

  1. Students must be capable of applying their knowledge to their work or vocation in a professional way and they should have building arguments and problem resolution skills within their area of study.
  2. Students must be capable of communicating information, ideas, problems and solutions to both specialised and non-specialised audiences.
  3. Actively demonstrate high concern for quality when defending or presenting the conclusions of one's work.
  4. Work in teams
  5. Apply the main methods for resolving ordinary differential equations and some simple partial derivative equations.
  6. Translate some real problems into the terms of ordinary differential equations and partial derivative equations.
  7. Resolve linear systems of ordinary differential equations.

Contents

1. First-Order Differential Equations in One Variable


1.1 Introduction to Differential Equations. Solution methods: separable equations, linear equations, exact equations, integrating factors. Change of variable.


1.2 Applications: Radioactive decay, mixture problems, population models, etc.



2. Fundamental Theorems


2.1 The space of continuous and bounded functions over a topological space: Existence and uniqueness of solutions, maximal interval of solutions, structure of the solution space, fundamental matrices.


2.2 Picard and Peano Theorems: Locally Lipschitz functions. Local existence and uniqueness. Stone-Weierstrass and Peano’s proof.


2.3 Extension of solutions: Existence and uniqueness of non-extendable solutions for problems with existence and uniqueness. Wintner's Lemma.


2.4 Continuous and differentiable dependence of solutions on initial conditions and parameters: Statement of theorems and examples.



3. Linear Equations


3.1 General properties of linear differential equations: Existence and uniqueness of solutions for the Cauchy problem, structure of the solution space for linear equations, fundamental matrices.


3.2 Systems of linear equations with constant coefficients: Matrix exponential. Computation of the exponential of Jordan canonical matrices. The nonhomogeneous case.


3.3 The linear equation of order n: General properties. Homogeneous equations with constant coefficients. Computation of particular solutions for the nonhomogeneous case.


3.4 The second-order linear equation: Mechanical systems, electrical circuits, forced periodic oscillations. The phenomenon of resonance.



4. Qualitative Theory of Autonomous Systems


4.1 Dynamical system induced by an autonomous differential equation. Critical points and periodic orbits. Stability. Equivalence and conjugacy.


4.2 Tubular flow theorem. Hartman’s theorem.


4.3 Qualitative study of linear equations.

Learning activities and methodology

Title Hours ECTS Learning outcomes
Theory classes 30 1.2
Study of the theory and resolution of problems 136 5.44
Problem classes 29 1.16
Practical modelization problems 16 0.64

Fundamental in the learning process of the subject is the work by the student, who can count on the guidance of the teacher at each moment.

There will be three types of guided activities:

Theory Classes: The teacher introduces the basic concepts of the subject matter showing examples, demostrating properties and fundamental results. The student must complement the teacher's explanations with personal study.

Classes of Problems: We work on the understanding and application of the concepts and tools introduced to theory, with the realization of theoretical and/or practical exercises. It is well known that the only way to learn mathematics is by solving lots and lots of problems. For this reason the student must dedicate a minimum of 5 hours a week to solving problems in this subject. The student will have a list of problems for each theme, which he must think about, try to solve and which will be worked on in the problem classes. A delivery of problems is requested for each theme to ensure that this work is done continuously.

Computer practices: in each session a different type of differential equation will be dealt with to model a real situation and predict future behaviors depending on circumstancial parameters.

The exercises that appear in the lists of Problems or Computer Practices and that have not finished in the corresponding session the student will have to solve them like part of his autonomous work.

The notes on the Theory, the lists of Problems and Computer Practices will be posted on the subject's Moodle Aules website; a summary of the Theory and Problem classes will be posted weekly.

Annotation: within the schedule set by the centre or degree programme, 15 minutes of one class will be reserved for students to evaluate their lecturers and their courses or modules through questionnaires.

Assessment

Continuous assessment activities

Title Weight Hours ECTS Learning outcomes
Resit exam 85% 4 0.16 1, 3, 5, 6, 7
Second partial exam 45% 4 0.16 1, 3, 5, 6, 7
First partial exam 40% 4 0.16 1, 3, 5, 6, 7
Examination of Modelling Practices 15% 2 0.08 1, 2, 4, 6, 7

Continuous assessment It will consist of the following assessment activities:

  • Modelling Practices Exam, with a weight of 15%. This activity is NOT recoverable.
  • A midterm exam at the middle of the semester, with a weight of 40%.
  • A final exam at the end of the semester, with a weight of 45%.


Resit: Weighted average of the grade obtained in a full subject exam (85%) and the grade obtained in the modelling practices exam (15%).


Single assessment Weighted average of the grade obtained in a full subject exam (85%) and the grade obtained in the modelling practices exam (15%). In case of failure, the same resit system as for continuous assessment will apply.


A student will be considered to obtain a Not Assessable grade if the weighted proportion of completed assessment activities is less than two thirds of those scheduled for the subject.


Note: In this subject, the use of Artificial Intelligence (AI) technologies is not permitted in any of its phases. Any work that includes AI-generated content will be considered an act of academic dishonesty and may result in a partial or total penalty on the activity grade, or more severe sanctions in serious cases.


Bibliography

P. Blanchard, and R.L. Devaney. Differential Equations. G.R. Hall, 2002. Traduït al castellà: "Ecuaciones Diferenciales". International Thomson Editores, México, 1999.

E. Boyce, y R.C. Di Prima. Ecuaciones Diferenciales y Problemas con Valores en la Frontera. Ed. Limusa, México, 1967.

R. Cubarsí. Equacions diferencials i la transformada de Laplace. Iniciativa Digital Politècnica, 2012. (http://hdl.handle.net/2099.3/36610)

C. Fernandez y J.M. Vegas. Ecuaciones diferenciales. Pirámide, Madrid, 1996.

G. Fulford, P. Forrester, A. Jones. Modelling with differential and difference equations. Cambridge University Press, New York, 1997.

M. Guzmán. Ecuaciones diferenciales ordinarias. Ed. Alhambra, Madrid, 1978.

V. Jimenez. Ecuaciones diferenciales. Serie: enseñanza. Universidad de Murcia, 2000.

M.C. Leseduarte, M. D. Llongueras, A. Magaña, R. Quintanilla de Latorre. Equacions Diferencials: Problemes resolts. Iniciativa Digital Politècnica, 2012. (http://hdl.handle.net/2099.3/36607)

F. Mañosas. Apunts d'Equacions diferencials. Campus virtual.

R. Martínez. Models amb Equacions Diferencials. Materials de la UAB, Servei de Publicacions de la UAB, no. 149. Bellaterra, 2004.

G.A. Muñozy J.B. Seoane. Fundamentos y problemas resueltos de Teoria cualitativa de ecuaciones diferenciales. Paraninfo, Madrid, 2017.

H. Ricardo. Ecuaciones diferenciales: una introducción moderna. Editorial Reverté, Barcelona, 2008.

J. Sotomayor. Lições de Equações Diferenciais Ordinárias. IMPA, 2025 (segona edició).

D.G. Zill. Ecuaciones diferenciales con aplicaciones de modelado. International Thomson Editores,México, 2001.

Software

Anything related to differential equations.

Course groups and languages

The information provided is provisional until November 30. After this date, you will be able to consult the language of each group through this link. To access the information, you will need to enter the course CODE

Type of teaching Group Language Semester Shift
(TE) Theory 1 Catalan first semester morning-mixed
(PAUL) Classroom practices 1 Catalan/Spanish first semester morning-mixed
(PLAB) Practical laboratories 1 Catalan/Spanish first semester morning-mixed
(PAUL) Classroom practices 2 Catalan/Spanish first semester morning-mixed
(PLAB) Practical laboratories 2 Catalan/Spanish first semester morning-mixed