# Differential Equations

## Course Description:

Study of differential equations. Qualitative theory and basic methods of solving differential equations and systems. Applications to modeling physical problems.

### Prerequisite Knowledge

Analysis I, Analysis II (Basic concepts for the last unit), Linear Algebra Εφαρμογές

### Course Units

# Title Description Hours
1 Introduction Derivation - mathematical models, concept of a differential equation, classification, concept of solution, initial boundary value problems, well-posed problems 2
2 First-order Differential Equations Linear equations of separated variables, exact differential equation and integrating factors, homogeneous, formulation of the theory of existence and uniqueness, modeling of physical problems. 8
3 Linear Differential Equations Theory of homogeneous differential equation, linear independence of functions or solutions and Wronskian determinant, Abel's theorem, reduction of the order - d'Alembert's method, non-homogeneous differential equation and the method of varying coefficients - Lagrange method, equations with constant coefficients, characteristic polynomial-simple, multiple, complex roots, method of determining coefficients. 8
4 Laplace Transform Definition, solving initial value problems, Heaviside and Dirac functions, equations with discontinuous non-homogeneous term, convolution theorem, Volterra equations 6
5 First Order Systems Differential Equation Homogeneous linear with constant coefficients, complex, multiple eigenvalues, phase portrait, autonomous systems and stability, non-homogeneous linear systems 6
6 Solving Linear Second Order using Power Series Solution in a regular point, Legendre equation, Legendre polynomials, Euler equation, solutions near a regular singular point, Bessel equation 6
7 Trigonometric Fourier Series Fourier-Euler coefficients, convergence theorem, even and odd functions -cos and sin expansions, complex form of Fourier series 4
8 BoundaryValue Problems Homogeneous Sturm-Liouville problems, eigenvalues and eigenfunctions 4
9 Separation of Variables Wave equation-vibrations of an elastic string, D'Alebert's solution. The method of separation of variables in two and three dimensions. Modeling of physical problems 8

### Learning Objectives

Upon successful completion of the course, students will be able to:

1. Know the important role of differential equations.
2. To have the ability to model using ordinary and partial differential equations.
3. Understand the importance of analytical and theoretical methods of problem solving

### Teaching Methods

 Teaching methods Lectures in class. Solving simple examples and problems in the classroom. Blackboard. Programming with Matlab. Yes. Students solve in the class with the help of tutors simple exercises using mainly Matlab in PC. Exercises Νο

### Student Assessment

• Final written exam: 70%
• Assignments (projects, reports): 30%

### Textbooks - Bibliography

1 W.BOYCE - R.DIPRIMA, ΣΤΟΙΧΕΙΩΔΕΙΣ ΔΙΑΦΟΡΙΚΕΣ ΕΞΙΣΩΣΕΙΣ ΚΑΙ ΠΡΟΒΛΗΜΑΤΑ ΣΥΝΟΡΙΑΚΩΝ ΤΙΜΩΝ, ΠΑΝ. ΕΚΔΟΣΕΙΣ ΕΜΠ, 2015 2 Ν.Δ.ΑΛΙΚΑΚΟΣ, ΚΑΛΟΓΕΡΟΠΟΥ-ΛΟΣ Γ.Η., ΣΥΝΗΘΕΙΣ ΔΙΑΦΟΡΙΚΕΣ ΕΞΙΣΩΣΕΙΣ, ΣΥΓΧΡΟΝΗ ΕΚΔΟΤΙΚΗ, 2003. 3 Ι. ΠΟΛΥΡΑΚΗΣ, ΣΥΝΗΘΕΙΣ ΔΙΑΦΟΡΙΚΕΣ ΕΞΙΣΩΣΕΙΣ, Ι. ΠΟΛΥΡΑΚΗΣ, 1998 4 Γ.Παντελίδης, Δ.Κραββαρίτης, Ν.Χατζησάββας, Συνήθεις Διαφορικές Εξισώσεις, Ζήτη, 1990

## Lecture Time - Place:

• Tuesday, 12:45 – 14:30,
Rooms:
• Αμφ. 1/2
• Friday, 10:45 – 13:30,
Rooms:
• Αμφ. 1/2

© 2017 School of Civil Engineering, ΕΜΠ