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  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  55000008Calculus II 55000002Algebra 55000001Calculus I 

Module  General information on the subject (55000001Calculus I) General information on the subject (55000008Calculus II) General information on the subject (55000002Algebra) 

REQUIRED SKILLS AND ABILITIES  


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 PoincaréBendixson 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  


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


X 


An ability to work on multidisciplinary teams.  
X 


X 


X 


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 lifelong 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 
TeleExerc 
Works 

60 
0 
0 
0 
0 
0 
0 
100 
0 
20 
0 
0 
0 
120 
X  Lectures 
Lab practice  
Projectbased 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):
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  