# Differential Equations, 2018

**ELENA ROMERA COLMENAREJO**

Departamento de Matemáticas

Universidad Carlos III de Madrid

Area:

Mathematics

Degree:

Degree in Biomedical Engineering

December 2018

Theorethical and Lab hours: 56

Total hours:90

**PRERREQUISITES AND RECOMMENDED PREVIOUS KNOWLEDGE**

It is necessary a basis on Differential and Integral Calculus of One and Several variables and also Linear Algebra.

**GENERAL DESCRIPTION OF THE SUBJECT**

The first part of the course is dedicated to Ordinary Differential Equations. It is divided into three chapters:

1. First Order Diffferential Equations, with the introduction to differential equations and the resolution with different methods and applications.

2.Linear Equations of HIgher Order, with the general study and the resolution of linear equations with constant and also some nonconstant coefficients.

3. Laplace Transform, a useful tool to solve differential equations and systems of differential equations and also integral equations.

The second part is dedicated to Partial Differential Equations and also has three chapters:

4. Method of Separation of Variables, with the application to the solution of initial and boundary value problems in partial differential equations. It also includes Fourier series.

5. Sturm-Liouville Problems. This chapter extends the previous method to the case of more general eigenvalue problems and studies generalized Fourier series.

6. Fourier Transform. It is a very useful method to solve differential equations on non-bounded domains.

**OBJETIVES: KNOWLEDGE AND SKILLS**

Modelization of physical problems and in other scientific areas in terms of differential equations. Resolution of a very wide variety of differential equations and its applications. Application of the Laplace transform to solve equations and linear systems. Study of the most important models of the Mathematical Physics: the Laplace equation, the heat equation and the wave equation. Use of techniques of separation of variables and of Fourier transform.

**TEACHING MATERIAL**

The course includes the Lecture Notes for each chapter, with many examples. Also there is a collection of problems for each chapter and the detailed resolution of the problems for each one of the chapters.

**ASSESSMENT ACTIVITIES OR PRACTICAL ASSIGNMENTS**

There are different kinds of evaluation activities: Selfevaluations to prepare the exams, partial exams and final exams, together with the solutions, so that it is possible to evaluate the results. Some exams from previous years are included.

Course Contents