Differenetial Equations

Basic applications from the theory of ordinary differential equations are presented throughout this course. These can help the students of the Department understand and use the tools and methods encountered in this very important field of Applied Mathematics.

Objectives

1. Familiarisation with the different kinds, forms, classes and mathematical techniques of Differential Equations (DEs) theory, as well as with the construction of these equations as regards physical and mechanical problems arising from the science of electrical engineering. 2. Knowledge of classical methods of solving basic forms of DEs appearing in literature and applied to electrical engineering oriented problems, classical mechanics and physics. Moreover, knowledge of construction and (or) choice of appropriate techniques for the treatment of more complex forms governing linear and non-linear problems. 3. Projection and explanation of the solutions onto the physical background of the problems, modelled by these equations.

Prerequisites

There are no prerequisite courses. It is however recommended that students should have a basic knowledge of differential and integral calculus, as well as of linear algebra.

Syllabus

Introduction, basic definitions – Ordinary differential equations – Nonlinear differential equations – Linearity and linearization – First order linear equations – Bernoulli and Riccati equations – Behaviour of solutions, reduction to separable variables form – Homogeneous equations – Complete equations, integral factors – Approximation methods – Vectors field – Envelop, singular points of the family of solutions – Parametric solutions, Lagrange, Clairaut, Abel equations – The existence and uniqueness theorem referring to first order equations – First integrals and general solutions of nonlinear second order equations - Homogeneous linear equations of the second order, Wronski determinant, fundamental solutions, relation with Riccati equation – Homogeneous equation with constant coefficients – No homogeneous linear second order equation, method of undetermined coefficients, method of variation of parameters – Case study: mechanical and electrical oscillations – Euler equations – Linear equations of higher order – Linear systems of differential equations of the first order – Theorem of existence and uniqueness – Fundamental matrix of solutions of an homogeneous autonomous system – Solution for a no homogeneous system – Stability, character of the origin – First integrals, phase space, phase trajectories – Nonlinear autonomous systems of the first order – Critical points – Linear approximation – Stability theorem – Limit cycles.