Course: 2019/2020

Differential Equations

(15537)

Students are expected to have completed

Calculus I, Cálculus II and Linear Algebra

Competences and skills that will be acquired and learning results. Further information on this link

The student will learn the basic topics of ordinary and partial differential equations:
1. Resolution of first order differential equations.
2. Resolution of higher order linear differential equations.
3. Use of Laplace transform to solve linear differential equations and systems.
4. Separation of variables in partial differential equations.
5. Fourier series and generalized Fourier series solutions.

Description of contents: programme

1. Differential equations of first order.
1.1. Definitions and examples.
1.2. Elementary methods of resolution.
1.3. Applications.
2. Higher order differential equations.
2.1. Linear differential equations of order n with constant coefficients.
2.2. Equations with variable coefficients: order reduction and equidimensional equations.
2.3. Relation between systems and linear equations.
3. Laplace transform.
3.1. Definition and properties.
3.2. Transforming and back-transforming.
3.3. Application to the resolution of linear equations and systems.
4. Method of separation of variables.
4.1. Initial and boundary problems. Examples of partial differential equations from Mathematical Physics.
4.2. Different kinds of equations and data.
4.3. Odd, even and periodic extensions of a function. Trigonometric Fourier series.
4.4. Resolution of equations by separation of variables and Fourier series.
4.5. Complex form of Fourier series.
5. Sturm-Liouville problems.
5.1. Sturm-Liouville problems and theorem.
5.2. Rayleigh's quotient. Minimization theorem.
5.3. Resolution of equations by separation of variables and generalized Fourier series.
5.4. Sturm-Liouville problems in several variables.

Learning activities and methodology

1.- Master classes.
2.- Problem classes.
3.- Selfevaluations.
4.- Partial controls.
5.- Final exam.
6.- Tutorials.

Assessment System

- % end-of-term-examination 60
- % of continuous assessment (assigments, laboratory, practicals...) 40

Basic Bibliography

- HABERMAN, R. . Elementary Applied Partial Differential Equations, 3rd. ed. . Prentice Hall. 1998
- SIMMONS, G. F. ; KRANTZ, S. G.. Differential Equations. Theory, Technique, and Practice. McGraw-Hill Companies, Inc. . 2007

Additional Bibliography

- BRANNAN, J.R. ; BOYCE, W.E.. Differential Equations with Boundary Value Problems: An Introduction to Modern Methods & Applications. Wiley.. 2010
- EDWARDS, C.H. PENNEY, D. E.. Elementary differential equations with boundary value problems por . Pearson Education2014.
- NAGLE, R. KENT; SAFF, E.B.; SNIDER A. D.. Fundamentals of differential equations . Pearson Addison-Wesley. 2008, 7th ed.
- SIMMONS, G. F.. Differential equations with applications and historical notes . CRC Press Textbooks in mathematics. 2017, 3rd edition