## Public University of Navarre

 Course code: 504209 Subject title: ORDINARY DIFFERENTIAL EQUATIONS Credits: 6 Type of subject: Mandatory Year: 2 Period: 1º S Department: Estadística, Informática y Matemáticas Lecturers: PALACIAN SUBIELA, JESUS FCO.   [Mentoring ] LOPEZ GARCIA, JOSE LUIS (Resp)   [Mentoring ]

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## ORDINARY DIFFERENTIAL EQUATIONS

Matemáticas/Ecuaciones Diferenciales y Álgebra

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### General proficiencies

CB4 - That students can transmit information, ideas, problems and solutions to both a specialized and non-specialized audience.

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### Specific proficiencies

CG1 - Apply acquired analytical and abstraction skills, intuition, and logical thinking to identify and analyze complex problems and seek and formulate solutions in a multidisciplinary environment.

CG5 - Prepare, plan and develop the contents of the topics and subjects of the scientific field corresponding to pre-university education.

CE19 - Propose and analyze mathematical models of real situations, applying the techniques of ordinary differential equations, partial differential equations, algebra and geometry to solve them.

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### Learning outcomes

RA 1. Master the concept of differential equation and system of differential equations, existence and uniqueness of solution.

RA 2. Know the basic techniques for solving first order differential equations.

RA 3. Understand the structure of the space of solutions of differential equations and linear systems. Master the basic techniques of solving differential equations and linear systems with constant coefficients.

RA 4. Handle the technique of solving linear differential equations using power series and its usefulness in the equations of mathematical physics.

RA 5. Get some knowledge concept of dynamical system and acquire the associated fundamental concepts.

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### Methodology

 Methodology - Activity Attendance Self-study A-1 Expository / participative classes 42 A-2 Hands on learning 14 A-3 Cooperative learning activities A-4 Carrying out tasks / group projects A-5 Studying and autonomous work of the student 88 A-6 Tutorials 2 A-7 Assessment Tests 4 Total 60 90

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### Relationship between formative activities and proficiencies/learning outcomes

 Proficiency Educational Activity CB4 - That students can transmit information, ideas, problems and solutions to both a specialized and non-specialized audience. A-1 Expository / participative classes; A-2 Hands on learning; A-5 Studying and autonomous work of the student; A-6 Tutorials and A-7 Assessment Tests. CG1 - Apply acquired analytical and abstraction skills, intuition, and logical thinking to identify and analyze complex problems and seek and formulate solutions in a multidisciplinary environment. A-1 Expository / participative classes; A-2 Hands on learning; A-5 Studying and autonomous work of the student; A-6 Tutorials and A-7 Assessment Tests. CG5 - Prepare, plan and develop the contents of the subjects and subjects of the scientific field corresponding to teachings. Prepare, plan and develop the contents of the subjects and subjects of the scientific field corresponding to pre-university teachings. A-1 Expository / participative classes; A-2 Hands on learning; A-5 Studying and autonomous work of the student; A-6 Tutorials and A-7 Assessment Tests. CE19 - Propose and analyze mathematical models of real situations, using the tools of ordinary differential equations, partial differential equations, algebra and geometry to solve them. A-1 Expository / participative classes; A-2 Hands on learning; A-5 Studying and autonomous work of the student; A-6 Tutorials and A-7 Assessment Tests.

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ENGLISH

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### Evaluation

 Learning Outcome Assessment System Weight (%) Can be retaken RA1-RA5 Tasks and Report: Continuous Assessment 20% no RA1-RA5 Individual written test (long answer tests) 80% yes

The ordinary call consists in four mid-course exams distributed homogenously along the course. Each exam will last an hour and have the same weight for the final mark. The extraordinary call consists in  a unique exam lasting 4 hours. Pocket-calculator is permitted in the exams.

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### Contents

First-order differential equations. Linear equations and power series solutions. Linear systems. Dynamical systems. Differential equations of physics. Sturm-Liouville problems.

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### Agenda

1. Introduction to differential equations. Motivation and basic definitions. Elementary differential equations.

2. First-order differential equations. Separation of variables. Homogeneous. Exact equations. Integrating factor. First-order linear. Changes of variables.

3. Theoretical aspects. Initial value problem. Existence and uniqueness theorem. Boundary problems. Approximate solutions.

4. Linear differential equations. General theory. Constant coefficients. Regular and singular points. Analytical solutions. Power series and Frobenius methods. Special functions. Introduction to Sturm-Liouville problems.

5. Linear systems of equations. General theory. Constant coefficients. Homogeneous and non-homogeneous systems.

6. Nonlinear equations. Introduction to dynamical systems. Applications in real life.

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### Bibliography

Access the bibliography that your professor has requested from the Library.

Basic bibliography

[1] D.G. Zill. A First Course in Differential Equations with Modeling Applications Problems, Tenth Ed., Brooks/Cole, Cengage Learning, 2013.

[1]  F. Diacu. An Introduction to Differential Equations: Order and Chaos. W.H. Freeman, 2000.

[2]  M.W. Hirsch,  S. Smale y R.L. Devaney. Differential Equations, Dynamical Systems, and an Introduction to Chaos. Academic Press, 2012.

[3] F. Marcellán, L. Casasús y A. Zarzo. Ecuaciones diferenciales. Problemas lineales y aplicaciones. McGraw-Hill, 1991.

[4] G. Strang. Differential Equations and Linear Algebra. Wellesley-Cambridge Press, 2015.

[5] F. Ayres. Schaum's Outline of Differential Equations. McGraw-Hill, 1992.

[6] Ó. Ciaurri. Instantáneas diferenciales. Métodos elementales de resolución de ecuaciones diferenciales ordinarias, estudio del problema de Cauchy y teoría de ecuaciones y sistemas lineales. Universidad de La Rioja, 2011.

[7] S. Novo, R. Obaya y J. Rojo. Ecuaciones y sistemas diferenciales. McGraw-Hill, 1995.

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### Location

Public University of Navarre. Aulario building (the room will be posted on the website). The specific places where each of the activities takes place will be announced at the beginning of the course.

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