Course: 2021/2022

Differential Equations

(15537)

Requirements (Subjects that are assumed to be known)

Calculus I, Cálculus II and Linear Algebra

SPECIFIC LEARNING OBJECTIVES (PO a):
- To understand the Linear operators and the principle of superposition for solving differential equations.
- To solve elementary differential equations by separation of variables and other methods.
- To understand the different application scopes of the differential equation to engineering and physics.
- To distinguish between elliptic, hyperbolic and parabolic partial differential equations and which initial or boundary conditions are appropriate for them.
- To understand how to apply separation of variables and the Fourier method to solve initial-boundary value problems for the equations of Mathematical Physics.
- To understand the separation of variables technique, the role of the resulting eigenvalue problems and the principle of superposition to solve initial-boundary value problems for the equations of Mathematical Physics.
- To understand when and how to use the method of characteristics to solve different cases of partial differential equations.
GENERAL ABILITIES (PO a, g, k):
- To understand the necessity of abstract thinking and formal mathematical proofs.
- To acquire communicative skills in mathematics.
- To acquire the ability to model real-world situations mathematically, with the aim of solving practical problems.
- To improve problem-solving skills.
(PO: a)

Description of contents: programme

1. Introduction
1.1 Basic models; direction fields
1.2 Classification of differential equations
2. First Order Differential Equations
2.1 Linear equations; integrating factors
2.2 Separable equations
2.3 Exact equations
3. Second Order Linear Equations
3.1 Definitions and examples
3.2 Linear homogeneous equations
3.3 Homogeneous equations with constant coefficients
3.4 Inhomogeneous equations: undetermined coefficients
3.5 Variation of constants
4. Systems of First Order Linear Equations
4.1 Basic theory; higher-order equations
4.2 Explicit solutions of non-homogeneous linear systems
4.3 Planar linear systems
5. Nonlinear Systems and Stability
5.1 Planar nonlinear systems
5.2 Stability
5.3 Periodic solutions
5.4 Higher-dimensional systems
6. Partial Differential Equations: Introduction
6.1 Examples and physical derivation
6.2 Types of equations and data; well- vs ill-posed problems
7. Separation of Variables
7.1 Problem resolution by separation of variables
7.2 Fourier trigonometric series: basic properties
8. Boundary-value Problems
8.1 Sturm-Liouville problems
8.2 Self-adjoint operators and spectrum
8.3 Rayleigh¿s quotient
8.4 Generalized Fourier series
8.5 Multivariable Sturm-Liouville problems
9. Non-Homogeneous Problems
9.1 Shifting the data
9.2 Fredholm¿s alternative
9.3 Eigenfunction expansions

Learning activities and methodology

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

Assessment System

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

Basic Bibliography

- Haberman, R.. Ecuaciones en derivadas parciales con series de Fourier y problemas de contorno. Prentice Hall. 2003
- Robinson, J. C.. An Introduction to Ordinary Differential Equations. Cambridge University Press. 2004
- Simmons, G. F. ; Krantz, S. G.. Ecuaciones diferenciales. Teoría, técnica y práctica. McGraw-Hill. 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., Calvis, D.. Differential Equations and Boundary Value Problems: Computing and Modeling. Pearson Education. 2016
- Nagle, R. K., Saff, E. B., Snider, A. D.. Fundamentals of differential equations . Pearson Addison-Wesley. 2008, 7th ed.
- Tikhonov, A. N., Samarskii, A. A.. Equations of Mathematical Physics. Dover. 1990

The course syllabus may change due academic events or other reasons.