Ordinary Differential Equations

Introduce students to the field of qualitative theory of Ordinary Differential Equations (ODE), both from a theoretical and practical point of view.

To show the interest of the qualitative study of the ODE since it allows to deduce the behavior of the solutions, without needing to solve the studied system. This resolution can be very complicate in linear systems of high dimension, being impossible in most non-linear systems.

Address the stability and instability of the critical points of the system, characterizing them in the linear case and in some non-linear systems. Obtain sufficient conditions that guarantee stability and instability in the rest of the non-linear systems.

Motivate the study with mathematical models that arise from different problems in different scientific fields

Represent the evolution of the orbits of the different systems using different computer programs.

1.- Autonomous systems in R^n. Properties. Motivations and examples. Phase portrait associated with a vector field. (4 hours exposition approximately)

2.- Stability and asymptotic stability for autonomous linear systems in R^n. Applications. (3 hours exposition approximately)

3.- Stability and asymptotic stability of solutions of non-linear autonomous systems. Application to problems of physics, biology or medicine:
3.1.- Method of the first approach. (3 hours exposition approximately)
3.2.- Lyapunov's method. Region of attraction. (4 hours exposition approximately)

Basic bibliography:

Bibliografía básica:

AGARWAL, R. P.; O’REGAN, D.; An Introduction to Ordinary Differential Equations. Springer, 2008.

DIZ PITA, É. Sistemas de ecuacións diferenciais autónomos en R^n. Unidades Didacticas USC. Editorial: Servicio de Publicacións e Intercambio Científico USC, 2022
https://dx.doi.org/10.15304/9788418445965 ISSN/ISBN 9788418445965

DIZ PITA, É.; OTERO ESPINAR, M. V.; Estabilidade de sistemas lineares. Unidades Didacticas USC. Editorial: Servicio de Publicacións e Intercambio Científico USC, 2022
https://dx.doi.org/10.15304/9788419155719, ISSN/ISBN 978-84-19155-71-9

DIZ PITA, É.; OTERO ESPINAR, M. V.; Estabilidade de sistemas non lineares. Unidades Didacticas USC. Servicio de Publicacións e Intercambio Científico USC, 2022
https://dx.doi.org/10.15304/9788419155702, ISSN/ISBN 978-84-19155-70-2


FERNANDEZ PÉREZ, C.; Ecuaciones Diferenciales I. Pirámide, 1992.

FERNANDEZ PÉREZ, C.; VEGAS MONTANER, J. M.; Ecuaciones Diferenciales II. Pirámide, 1996.

LOGEMANN, H.; RYAN, E.; Ordinary Differential Equations. Analysis, Qualitative Theory and Control. Springer, 2014.

PRECUP, R.; Ordinary Differential Equations. De Gruyter, 2018.

Complementary bibliography:

BRAUER, F.; NOHEL, J. A.; Qualitative Theory of Ordinary Differential Equations. Benjamin, 1969.

BRAUM, M.; Ecuaciones Diferenciales y sus Aplicaciones, Grupo Editorial Iberoamérica, 1990.

CODDINGTON, E. A.; LEVINSON, N.; Theory of Ordinary Differential Equations. McGraw-Hill, 1955.

CRONIN, J.; Ordinary Differential Equations. Introduction and Qualitative Theory. Chapman & Hall, 2008.

EDWARDWS, C. H.; PENNEY, D. E.; Ecuaciones Diferenciales. Prentice Hall, 2001.

GODUNOV, S. K.; Ordinary Differential Equations with Constant Coefficient. American Mathematical Society, 1997.

GUZMÁN, M.; Ecuaciones Diferenciales Ordinarias. Teoría de Estabilidad y Control. Alhambra, 1987.

HIRSCH, M. W.; SMALE, S.; Ecuaciones Diferenciales, Sistemas Dinámicos y Álgebra Lineal. Alianza Universidad, 1983.

JORDAN, D.W.; SMITH, P.; Nonlinear Ordinary Differential Equations. Oxford Univ. Press, 1999.

KRANTZ, S.; Differential Equations. Theory, Technique, and Practice with Boundary Value Poblems. CRC Press, 2016.

OTERO ESPINAR, M. V.; DIZ PITA, É.; Ecuacións diferenciais ordinarias: modelos esenciais aplicados a distintos ámbitos científicos. Esenciais USC. Servicio de Publicacións e Intercambio Científico USC, 2021. ISSN/ISBN 9788418445781

PEREZ GARCÍA, V. M., Problemas de Ecuaciones Diferenciales. Ariel, 2001.

SIMMONS, G. F.; Ecuaciones Diferenciales. McGraw-Hill. 1993.

SOTOMAYOR, J.; Liçoes de Equaçoes Diferenciais Ordinarias. I.M.P.A., 1979.

In this subject we will try to contribute that the students attain the following competences collected in the memory of the Degree in Mathematics of the USC.

General skills
CG1 - Know the most important concepts, methods and results of the different branches of Mathematics, together with a certain historical perspective of their development
CG2 - Collect and interpret data, information and relevant results, obtain conclusions and issue reasoned reports on scientific, technological or other problems that require the use of mathematical tools.
CG3 - Apply both acquired theoretical and practical knowledge as well as the capacity for analysis and abstraction in the definition and approach of problems and in the search for their solutions both in academic and professional contexts
CG4 - Communicate, both in writing and orally, knowledge, procedures, results and ideas in Mathematics for both a specialized and non-specialized audience.
CG5 - Study and learn autonomously, with organization of time and resources, new knowledge and techniques in any scientific or technological discipline.

Specific competences
CE1 - Understand and use mathematical language
CE2 - Know rigorous proofs of some classical theorems in different areas of Mathematics.
CE3 - Devise demonstrations of mathematical results, formulate conjectures and imagine strategies to confirm or deny them.
CE4 - Identify errors in incorrect reasoning proposing demonstrations or counterexamples.
CE5 - Assimilate the definition of a new mathematical object, relate it to others already known, and be able to use it in different contexts
CE6 - Knowing how to abstract the properties and substantial facts of a problem, distinguishing them from those that are purely occasional or circumstantial
CE7 - Propose, analyze, validate and interpret models of simple real situations, using the most appropriate mathematical tools for the purposes pursued
CE8 - Plan and execute algorithms and mathematical methods to solve problems in the academic, technical, financial or social field
CE9 - Use computer applications of statistical analysis, numerical and symbolic calculation, graphical visualization, optimization and scientific software, in general, to experiment in Mathematics and solve problems

Transversal competences
CT1 - Use bibliography and search tools for general and specific bibliographical resources of Mathematics, including Internet access
CT2 - Manage optimal working time and organize available resources, establishing priorities, alternative paths and identifying logical errors in decision making
CT3 - Check or reasonably refute the arguments of other people
CT5 - Read scientific texts both in their own language and in others of relevance in the scientific field, especially English.

In addition, the training activities to be developed have the objective that students acquire the following competences and learning outcomes related to the Differential Equations module:
Understand, learn and know how to express rigorously the concepts and techniques developed in the program.
To extract qualitative information, without their explicit resolution, of the solutions of an ordinary differential equation.
Know the characterization of the stability of linear systems.
Know techniques for studying the stability of non-linear systems.
Know relevant examples of differential equations of physics and other sciences.
Translate in terms of ordinary differential equations some problems of the applied sciences (physics, chemistry, biology, medicine, etc.).
Use the studied mathematical techniques, to understand the dynamics of the obtained equations (phase portrait, stability, etc.) and interpret the results, evaluating the strengths and weaknesses of the proposed model.

We will work in a special way: The rigorous and clear expression, both oral and written, the logical reasoning and identification of errors in the procedures, the capacity of abstraction, creativity, team work, the development of the analytical capacity in solving problems and critical attitude with different solutions.

The general methodological indications established in the Memory of the Degree in Mathematics of the University of Santiago de Compostela (USC) will be followed.

Teaching is scheduled in expository and interactive classes, some of which will consist of computer practices.

Expository Teaching (14 hours): The expository classes will be dedicated to the presentation and development of the essential contents of the subject.

Seminary and laboratory teaching (28 hours): The interactive seminary classes (14 hours) will be dedicated to the presentation of examples and problem solving (both theoretical and in the field of applications). In the interactive laboratory classes (14 hours) individual or group work will be organized and problems will be proposed to be solved by the students. In these classes, the discussion and debate with the students, as well as the resolution and exposition on their part of the proposed tasks, is fundamental so that the knowledge is practiced and strengthened and some of the mentioned competences are worked on. Students will use computer packages of symbolic calculation related to the contents of the subject.

Tutorials (2 hours): The tutoring sessions are designed to stimulate student activity outside of class. These tutorials in very small groups will serve so that the students can examine their learning process at all times.

All the training activities carried out in each group will be coordinated to guarantee the training equivalence of all the groups.

In a generic way, either the USC virtual course or the Teams platform will be used as a mechanism to provide students with the necessary resources for the development of the subject (notes, exercises, bulletins, etc.).

The expository and interactive teaching will be face-to-face and will be complemented with the USC virtual course or the platform MS Teams, in which the students will find bibliographic materials, bulletins of problems, and other didactic materials.

Tutorials will be face-to-face, although they can be carried out partially via email or via MS Teams.

For the calculation of the final mark, the grade of the continuous assessment and the grade of the final exam will be taken into account.

Continuous assessment will consist of one or two intermediate tasks, which will consist of a project work throughout the course or mid-term exams on dates announced well in advance and coordinated with the rest of the course subjects.

Through the different proposed activities will be evaluated, of course contextualizing the subject in 3º degree course, the acquisition of skills, such as CB2, CB3, CB4, CG2, CG3, CG4, CT1, CT2, CT3, CE7, CE8, CE9 or the ability to work in a team and to learn independently. The qualification obtained in the continuous evaluation will be applied in the two opportunities of a same academic course (first semester and July).

A final written exam will be carried out, which will allow to verify the knowledge acquired in relation to the concepts and results of the subject and the capacity of its application to concrete cases, both theoretical and applied. The written exam will consist of theoretical or practical questions, in addition to the specific competences of the subject, the competences CB1, CB2, CB4, CB5, CG1, CG3, CG4, CE1, CE2, CE3, CE4, CE5, CE6 will be evaluated.

Although they will not be the same in the different groups, both the continuous assessment activities and the final exam will be coordinated to guarantee the formative equivalence of all the groups.

With the mark of the continuous formative evaluation (C) and the mark of the final written exam (F), the final mark in the subject (NF) will be calculated according to the following formula:
NF = max {F, 0.4 * C + 0.6 * F}

It will be understood as NOT PRESENTED who does not make the final written exam.

On the second opportunity, the same evaluation system will be used, but with the test corresponding to the second opportunity, which will be a written final exam of the same type as the first.

Continuous assessment tasks may be performed and/or delivered in person or through the USC virtual campus or MS Teams as appropriate in each case.

The final exam will be held in person.

CLASSROOM WORK
Expositive classes (14 h)
Seminar classes (14 h)
Laboratory classes (14 h)
Tutoring in very small groups (2 h)

Total work hours in the classroom: 44

PERSONAL STUDENT WORK
Self-study individual or group (42 hours)
Writing of exercises, conclusions and other works (15 h)
Programming/experimentation and other computer work (7.5 h)
Suggested readings, activities in library or similar (5h)

Total hours student's personal work: 68.5

TOTAL: 112,5 hours

The students must handle with ease the subjects studied in the subjects "Introduction to Mathematical Analysis", "Continuity and Differentiability of One Real Variable Functions", "Integration of One Real Variable Functions", "Differentiation of Functions of Various Real Variables" and " Introduction to Ordinary Differential Equations".

Starting from this situation, they should work regularly (daily) and rigorously. It is essential to actively participate in the learning process of the subject, attend regularly to both theoretical and practical classes in a participatory manner, especially in classes in small groups and tutorials, asking the relevant questions.