The student will learn the basic topics of ordinary and partial differential equations:
1. Solving first order differential equations.
2. Solving higher order, linear differential equations.
3. Using Laplace transform to solve linear differential equations and systems.
4. Separation of variables in partial differential equations.
5. Finding solutions as Fourier series and generalized Fourier series.
Description of contents: programme
1. First Order Differential Equations.
a. Definitions and examples.
b. Elementary resolution methods.
c. Applications.
2. Higher Order Differential Equations.
a. Linear equations of order n with constant coefficients.
b. Equations with variable coefficientes: order reduction and equidimensional equations.
c. Relation between systems and linear equations.
3. Laplace Transform.
a. Definition and properties.
b. Transforming and anti-transforming.
c. Application to the resolution of linear differential equations and systems.
4. Introduction to Partial Differential Equations.
a. Initial and boundary values problems.
b. Examples of PDEs of Mathematical Physics.
c. Different kinds of equations and data.
d. Classification of second order linear PDE's.
5. Method of separation of variables.
a. Even, odd, and periodic extensiones of a function. Trigonometric Fourier series.
b. Solving PDE's using separation of variables and Fourier series.
c. Complex form of Fourier series.
6. Sturm-Liouville Problems.
a. Self-adjoint Sturm-Liouville problems.
b. Rayleigh's quotient. Minimization theorem.
c. Solving PDEs using separation of variables and generalized Fourier series.
d. Sturm-Liouville problems in several variables.
Learning activities and methodology
1.- Master classes, in which the theroretical concepts are presented, together with examples.
2.- Problem classes, to state and solve the proposed exercises.
3.- Selfevaluations.
4.- Partial controls.
5.- Final examination.
6.- Tutorials.
Assessment System
% end-of-term-examination 60
% of continuous assessment (assigments, laboratory, practicals...) 40
Basic Bibliography
D.G. ZILL. Ecuaciones Diferenciales con Aplicaciones de Modelado, sexta edición. Thomson. 1997