ORDINARY DIFFERENTIAL EQUATIONS
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- Versione italiana
- Academic year
- 2022/2023
- Teacher
- DAMIANO FOSCHI
- 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]
* Existence for the Cauchy problem in the case of continuity. Ascoli-Arzelà Ascoli theorem. Peano's theorem. [5 hours]
* First order linear systems: exponential of a matrix and Jordan normal form. [8 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 lectures of theory, applications and exercises.
Learning assessment procedures
- The course learning assessment is done through homework assignments and a final exam. Admission to the final exam is granted only after submission of the homework assignments.
The exam consists of two parts:
- a written examination of 3 hours, where the student is asked to solve some exercises (usually 4)
- an oral examination, consisting in a discussion about the theoretical arguments of the course.
The student is asked to discuss some of the theorems studied and of their applications.
A student is admitted to the oral examination if his mark at the written examination is greater or equal to 15. The oral examination has to be taken in the sames session as the written exam, and before the beginning of the next term's lessons.
The final mark depends on both the examinations. It isn't given by the arithmetical mean between the written and the oral mark, but it comes from an overall evaluation of the student's skill.
During the summer session there will be at least 2 possibilities to take both the written and the oral exam, the dates will be arranged together with the students of the class. In the other sessions there will be one or more opportunities to take the exam, depending on the student's demand. 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.
