ORDINARY DIFFERENTIAL EQUATIONS
Academic year and teacher
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- Versione italiana
- Academic year
- 2019/2020
- Teacher
- FRANCESCA AGNESE PRINARI
- Credits
- 6
- Didactic period
- Secondo Semestre
- SSD
- MAT/05
Training objectives
- The main educational goals of the course are:
* To introduce the student to the concepts and key issues related to ordinary differential equations.
* To provide methods for solving equations and systems of linear and nonlinear differential equations.
* To provide tools for the qualitative study of the solutions of a differential equation.
The main acquired knowledge will be:
* concrete examples of models of ordinary differential equations;
* classification and examples of the different types of ordinary differential equations;
* study of separable variables differential equations, or which can be reduced to separable variables equations;
* study of linear differential equations and systems with constant coefficients;
* exponential of a matrix;
* basic general theory: uniqueness and existence theoremts for local solutions of the Cauchy problem;
* extension of solutions and behaviour of maximal solutions;
* differential inequalities, Gronwall's lemma and comparison theorems;
* qualitative analysis of solutions of scalar equations.
The basic acquired abilities (that are the capacity of applying the acquired knowledge) will be to be able to:
* set up a sistem of differential equations starting from the description of the problem's physical model;
* transform a systen of ordinary differential equations into a system of first order and/or in normal form;
* solve separable variables equations, or scalar equations which can be reduced to separable variables ones;
* solve homogeneous and non homogeneous first order linear scalar differential equations with variable coefficients;
* solve homogeneous and non homogeneous linear scalar differential equations with constant coefficients;
* solve homogeneous and non homogeneous first order systems with constant coefficients;
* applya the reduction method to lower the order of a linear equation or the dimension of a linear system given one of its solutions;
* compute the exponenatial of a matrix;
* prove, under appropriate assumptions and with different methods, the existence and the uniqueness of solutions of the Cauchy problem;
* classify maximal solutions of a differential equation according to their behaviour at the boundary;
* draw the qualitative graph of solutions to scalar differential equations without solving explicitely the equation. Prerequisites
- Differential and integral calculus for functions of one and of several real variables.
Basic concepts of matrix linear algebra.
Computation of n-roots of a complex number.
Cartesian representation of graphs and curves. Course programme
- * Motivation and examples. Basic definitions (ordinary differential equations, Cauchy problem). Basic general theory: local existence and uniqueness of the solution under Lipschitz hypothesis by the method of Picard's iterations. [7 hours]
* Gronwall's Lemma. Maximal solutions and extension of solutions. Continuous dependence on the initial datum.Maximal solutions and extension of solutions. [7 hours]
* Basic resolution methods for differential equations in normal form: separable variables equations, first order linear differential equation, 1-forms and differential equations. Bernoulli's equation. Differential equations in a non-normal form. Clairaut's equation. Autonomous equations. Homogeneous equations. Riccati equation. Discussion and resolution of Cauchy problems related to various types of differential equations. [7 hours]
*Qualitative study of an equation: comparison theorem, monotonicity theorem and asymptote theorem. [7 hours]
*Weirstrass discussion. [2 Hours]
* Existence for the Cauchy problem in the case of continuity. Ascoli-Arzelà Ascoli theorem. Peano's theorem. [4 hours]
* First order linear systems: exponential of a matrix and Jordan normal form. [7 hours]
*Linear differential equations of order n. Characteristic polynomial. Euler equation. Method of variation of constants. D'Alembert method for the reduction of the order. [7 hours] Didactic methods
- * 48 hours of class lectures of theory, applications and exercises.
Learning assessment procedures
- The examinations is divided in two parts: a written partial exam (which requires the resolution of some exercises) and an oral examination. To pass the written test it is required to get at least 16 points out of 31. The final mark V takes into account both the partial scores according to the formula:
V=(2m+3 M):5+1, rounded up to the nearest integer,
where m=minimum{O,S}, M=maximum{O,S} and O=mark of the oral test and S=mark of the written test.
It is possibile to split the written partial exams in two parts. The first test takes place after the first half part of the course, the second one at the end of the course. After passing the written test, the student can take the oral test within the first data of the next session of exams. Reference texts
(1) Analisi matematica 2, by N.Fusco - P.Marcellini - C.Sbordone
(2) Introduzione alle equazioni differenziali ordinarie, by A. Malusa (Edizioni La Dotta)
(3) Equazioni differenziali ordinarie by L.Piccinnini - G.Stampacchia - G.Vidossich
(4)Analisi 2 by Bramanti-Pagani-Salsa
(5) Ordinary Differential Equations, by V.I.Arnold
(6) Lecture notes.
