The course aims to provide students with the preliminary language and basic concepts of differential equations and dynamical systems.
Course Prerequisites
The mathematics courses of the bachelor's degree program; in particular: differential and integral calculus for real functions, complex numbers, vector and matrix calculus, Linear differential equations with constant coefficients of first and second order.
Teaching Methods
Lectures and exercises.
Assessment Methods
The exam will be written. Students will have one hour. The written exam consists of two questions. The written test for Advanced Mathematical Methods for Engineers takes place simultaneously with the written test for Numerical Methods in Engineering Sciences, as well as with the optional oral exam. The results of both written parts determine the final grade for [510810] – ADVANCED MATHEMATICAL AND NUMERICAL METHODS FOR ENGINEERS. The oral exam is not mandatory. However, students who have obtained a positive overall grade (Adv. Math. Meth. + Num. Meth.) in the written part (i.e., at least 18/30) may choose to take an oral exam. The oral exam covers the topics presented during the lectures of both courses: Advanced Mathematical Methods for Engineers and Numerical Methods in Engineering Sciences. The oral exam can change the final grade in either direction: a poor performance may result in a failed exam. For students who have chosen the basic written exam for the Numerical Methods in Engineering Sciences part, the maximum grade achievable cannot exceed 24/30.
Texts
M.W. Hirsch, S. Smale. Differential Equations, Dynamical Systems and Linear Algebra. Academic Press, 1974. C.D. Pagani, S. Salsa, Analisi Matematica, Volume 2, Zanichelli, 2006 (Italian). H. Ricardo. A modern introduction to differential equations. Elsevier.
Contents
Ordinary differential equations: Basic definitions, examples and properties. First order linear equations and separation of variable method. The Cauchy problem. Existence and uniqueness: the Peano's theorem, the Cauchy-Lipschitz theorem. The Bernoulli and homogeneous equations. Qualitative study of solutions of Cauchy problems. Linear systems, exponential matrix, higher order linear ODEs with constant coefficients. Boundary value problems. Asymptotic behaviour and stability of dynamical systems. Examples. The linearization method.
