SKILL

To apply the main methods to solve ordinary differential equations.

To understand and precisely state the main concepts and results of the theory of existence and uniqueness of solutions to differential equations, using results of Mathematical Analysis previously studied. Also results regarding the dependance with respect to initial conditions.

To know rigorous proofs of the results concerning differential equations and propose new proofs of proposed results.

To use analytic, graphic and computational methods to solve certain differential equations.

To solve linear systems of ordinary differential equations.

To relate different problems from Geometry, Physics and the real world to differential equations.

To deduce qualitative information about the solutions to an ordinary differential equations, without solving it.

To solve differential equations and explain the process, orally or in written form, using the convenient mathematical language.

To translate real word problems in terms of ordinary differential equations or partial differential equations.

To understand the behavior of differential equations around regular or singular points, and the concept of stability of equilibria.



LEARNING OUTCOMES

To apply the main methods to solve differential equations, both ordinary or partial differential equations.

To solve linear systems of ordinary differential equations.

To understand real world problems in terms of differential equations.

To learn qualitative information about solutions to differential equations.

1. INTRODUCTION: definitions, the concept of solution, classification, geometric description of the solutions, family of orthogonal trajectories or curves, problems from science and technology.

2. ELEMENTARY METHODS: equations of separable variables, homogeneous equations, linear equations, Bernoulli equation, Ricatti equations, exact equations, integrating factors, second order equations that reduce to two equations of order one.

3. LINEAR EQUATIONS: homogeneous equations, Liouville formula, order reduction, non-homogeneous equations: variation of constants, constant coefficient equations, Euler equations.

4. SOLUTIONS IN SERIES: regular points, regular singular points; indicial equation: real simple roots whose difference is non-integer, real simple roots whose difference is an integer, double real root; Bessel functions.

5. LINEAR SYSTEMS: homogeneous systems, fundamental matrix, Jacobi formula, constant coefficients systems, reduction method, the matrix exponential, the method of eigenvectors.

6. EXISTENCE THEORY: the Cauchy problem, Lipschitz condition, Picard approximations, existence and uniqueness of solution, interval of existence, dependence on inicial conditions and parameters.

7. AUTONOMOUS SYSTEMS: the phase plane, orbits, critical points, stability and asymptotic stability; stability of linear systems, classification of critical points; nonlinear systems: stability by linealization, conservative systems, Poincaré and Liapunov theorems.

8. STURM-LIOUVILLE PROBLEMS: Fouriser series of a function, Fourier series with respect to and orthogonal system, pointwise convergence and convergence in L^2. Sturm-Liouville problem, eigenvalues and eigenfunctions, existence of eigenvalues, orthogonality of eigenfunctions, non-homogeneous Sturm-Liouville problems; Green's function.

9. INTRODUCTION TO PARTIAL DIFFERENTIAL EQUATIONS: First order partial differential equations. Existence of solution. Method of characteristics. Constant coefficients second order partial differential equations. Classification. Reduction to its canonical form. Method of characteristics. Solution to hyperbolic equations in a half-plane, in a quadrant.

10. SEPARATION OF VARIABLES: Solution to the vibrating string through separation of variables. Solution to the temperature distribution of a finite bar and of a circular plate through separation of variables. Solution to Laplace's equation in a rectangle and in a circular domain through separation of variables.

METHODOLOGY

The theoretical part of the course will be exposed in lectures, following the main references in the Bibliography. These lectures will be complemented with problem sessions where the students will answer questions applying the knowledge from the lectures.

In the seminars problems or examples related to the content of the course will be developed by the students, generally giving the problems previously so that they work on their own and discuss about them in the classroom.

The students will expose, individually or in groups, problems or part of the theory of the course. They will have the support of the professors to help the students in the resolution and exposition. The main part of the student's job is personal. The professors will guide them during their office hours about any question or difficulty the students may have along the course.

Written exams, both theory and problems

Weight: 85%-100%

Criteria:

- Correct argumentation of the solutions and use of the definitions.

- Correct use of mathematical language.

- Clear and ordered arguments, indicating each step correctly.

- Exact results in the problems.

-Métodos de argumentación claros y ordenados explicando los pasos.





Trabajos de los seminarios (escritos y orales).

Peso: 0%-15%

Criterios:

-Respuestas correctas y buena utilización del lenguaje matemático

-Claridad en los razonamientos

-En las explicaciones orales orden y precisión

-Orden y precisión en la resolución de problemas

-Asistencia



La renuncia a la evaluacion continua se podra realizar hasta la semana 18 del curso, mediante escrito al responsable de la asignatura.

La evaluacion final consistira en un examen de toda la asignatura. Peso 100%.

eGela platform, if available.

Basic bibliography

BIBLIOGRAFÍA

*BOYCE-DIPRIMA, Ecuaciones diferenciales y problemas con valores en la frontera, Limusa.

*A. DOU, Ecuaciones en derivadas parciales, Dossat.

*KISELIOV, KRASNOV Y MAKARENKO, Problemas de ecuaciones diferenciales ordinarias, MIR.

*R. K. NAGGLE Y E. B. SAFF, Fundamentos de Ecuaciones Diferenciales, Addison-Wesley Iberoamericana, 1992.

*II. PERAL ALONSO, Primer curso de ecuaciones en derivadas parciales, Addison-Wesley/Universidad Autónoma de Madrid,1995.

*F. SIMMONS, Ecuaciones Diferenciales con Aplicaciones y Notas Históricas, McGraw Hill.

In-depth bibliography

*M. BRAUN, Differential Equations and Their Applications, Springer Verlag, New York 1978.
*M. W. HIRSCH, S. SMALE, Ecuaciones diferenciales, sistemas dinámicos y álgebra lineal, Alianza Editorial, Alianza Universidad, Textos nº 61.