ETSII Code 55000011 Name Differential Equations
Type of subject Common Official Degree Bachelor on Industrial Engineering Technologies
Department Mathematics Applied to Industrial Engineering Telephone  + 34 91 336 3018
Teaching unit Mathematics Web dmaii.etsii.upm.es
Thematic block E-mail matemat@etsii.upm.es
Language Semester
Field of specialisation
Head of the subject
1
 No specialised area Eva María Sánchez
Number of students
Academic year
 Classes/Sem Study factor
ECTS
Min
Max
 2nd year (Bachelor in Industrial Engineering Technologies)


REQUIRED KNOWLEDGE
Subject 55000008-Calculus II
55000002-Algebra
55000001-Calculus I
Module General information on the subject (55000001-Calculus I)
General information on the subject (55000008-Calculus II)
General information on the subject (55000002-Algebra)
REQUIRED SKILLS AND ABILITIES

  • A good command of mathematical language. 
  • A knowledge of the rules of mathematical logic: implication, equivalence, necessary or sufficient condition, etc.                   
  • An ability to make calculations with ease. 
  • A skill in the use of mathematical tools: a) Elementary techniques of Calculus: derivatives, chain rule, calculation of antiderivatives. b) Elementary techniques of Linear Alegbra: matrix calculus, diagonalization, eigenvalues and eigenvectors. c) Techniques of multivariable real calculus: differentiation, multiple and line integrals. d) Elementary handling of complex numbers: exponentials, plotting. 
  • Basic concepts of General Physics: velocity, acceleration, force fields, etc. 
  • Study and concentration skills.













SHORT TABLE OF CONTENTS KNOWLEDGE PROVIDED
MODULE 0:   General information on the subject Unit 0:   General information on the subject
MODULE 1:   Introduction to ODEs (Ordinary Differential Equations) Unit 1:   Formulation and solution to initial value and boundary value problems.
Unit 2:   Elementary methods  for solving ODEs: exact, separable, linear, homogeneous, and Bernouilli equations.
MODULE 2:   Linear  methods Unit 3: First order linear differential systems. Exponential of a matrix. Nonhomogeneous linear systems.
Unit 4: Higher order linear ODEs. Fundamental system of solutions. Nonhomogeneous case.
Unit 5:  Notions on ODEs and linear differential systems with variable coefficients: Change of variables. Order reduction. Euler's equation.
Unit 6: Plane linear systems: nodes, spirals, and centers.
MODULE 3: Nonlinear differential systems Unit 7:  Autonomous nonlinear differential systems. Phase space, orbits.
Unit 8:  Stability of equilibria: linearization and Lyapunov's direct method.
Unit 9:  Periodic solutions: Poincaré, Bendixson, and Poincaré-Bendixson theorems.
Unit 10:  Applications: models in ecology, electricity, economy, etc.
MODULE 4: Partial differential equations (PDE) Unit 11:   Notions on PDEs. Problems of  Mathematical Physics: wave equation, Laplace equation, and heat equation.
Unit 12:  Solving PDEs by the separation of variables method.
SKILLS AND ABILITIES PROVIDED

  • A capacity for abstraction and general concepts recognition in practical situations.
  • Provides a diverse range of tools to address the treatment of natural processes.
  • A skill to formulate and analyse models of natural processes. A skill to interpret the results obtained and evaluate the models that have been used.
  • An ability to apply analytical methods for solving technical problems known from other subjects.
  • Provides a wide panorama of classical models applied in very different fields: mechanics, theoretical ecology, economy, epidemiology, etc.


    GENERIC AND TRANSVERSAL COMPETENCES  TO WHICH IT CONTRIBUTES
    X   An ability to apply knowledge of mathematics, science, and engineering.
    X  

    An ability to design and conduct experiments, as well as to analyze and interpret data.
    X  

    An ability to design a system, component, or process to meet desired needs within realistic constraints such as economic, environmental, social, political, ethical, health and safety, manufacturability, and sustainability.
      An ability to work on multidisciplinary teams.
    X  

    An ability to identify, formulate, and solve engineering problems.  
    X  

    An understanding of professional and ethical responsability.
     X

    An ability to communicate effectively.
     X

    The broad education necessary to understand the impact of engineering solutions in a global, economic, environmental, and societal context.

     X A recognition of the need for, and an ability to engage in life-long learning.
     X

    A knowledge of contemporary issues.
    TEACHING METHODOLOGY
    Activities scheduled in the POD
    (Project of Teaching Organization)
    Other activities
    Total Teaching
    Personal study
    Total Study
    Conventional classroom Computer classroom Cooperative classroom Laboratory Practical classes

    Contents
    Practice
    Activi.
    Deliverables
    Tele-Exerc
    Works
    60
    0
    0
    0
    0
    0
    0
    100
    0
    20
    0
    0
    0
    120
    X   Lectures
      Lab practice
      Project-based practices
    X   Others: Practical classes: solving exercises and stating and solving problems.
    ASSESSMENT OF KNOWLEDGE
    A midterm exam representing 40% of the final score, and a comprehensive exam representing 60% of the final score.
    Continuous assessment.
        Types of tests and weight in the final score (higher than 35% advisable):
    • 40 % written tests.
    • 0 % periodic exercises.
    • 0% individual or group work.
    • 0 % self-assessment (AulaWeb, Mecfunnet).
    • 0 % oral submissions in public.
    • 0 % practice.
    • 60 % others (specify):  comprehensive exam.
    Final examination
       Minimum score required in final examination: 5
    ASSESSMENT OF SKILLS AND ABILITIES
     
    ASSESSMENT OF GENERIC COMPETENCES
     
    BIBLIOGRAFY

    E. Sánchez, J. González y J. Gutiérrez. Sistemas dinámicos. Una introducción a través de ejercicios. Sección de Publicaciones de la E.T.S.I. Industriales de la U.P.M.

    D.G. Zill y M.R. Culleu. Ecuaciones Diferenciales con problemas de valores en la frontera.
    RESOURCES

    ADDITIONAL INFORMATION