85169 - Differential Equations

Learning outcomes

By the end of the course the student knows the fundamentals of the theory of ordinary differential equations, including existence and uniqueness; the major results in the theory of stability; the class of differential equations solvable in closed form; the change of variable method and the Lie symmetry theory; the family of linear equations of second order with variable coefficients and their connections with the special functions theory.

Course contents

Module 1: prof Ritelli

Solvable differential equations by means of elementary techniques:
a) Equation of first order: separable equations, homogeneous equations, linear equations, exact equations, integrating factor, Bernoulli equation, Riccati equation.
b) Equation of second and higher order: linear homogeneous differential equations with constant coefficients, non homogeneous equation, variation of parameters.
c) System of linear differential equations with constant coefficients

Linear second differential equation with variable coefficients
a) Preliminary to the series solution
b) Solution at an ordinary point
c) Solution at a singular point
d) Hypergeometric equation
e) Bessel equation

Module 2: prof. Bernardi

Existence and Uniqueness with the Lipschitz condition

Existence without the Lipschitz condition

Some global properties of solutions

Analytic differential equations

Dependance on data

Nonuniqueness

The Wiener process

The Stochastic Integral

Stochastic Differential Equations of First Order

Readings/Bibliography

Module 1

Lecture notes prepared by the instructor.

Ravi P. Agarwal, Donal O'Reagan: Ordinary and Partial Differential Equation. Springer 2009 (Chapters 1-10)

Module 2

Stochastic Differential Equations
I.I.Gihman, A.V. Skorohod
Springer 1972

Basic Theory of Ordinary Differential Equations
Po-Fang Hsieh
Yasutaka Sibuya
Springer 1999

 

Teaching methods

Module 1

Lessons ex Cathedra: 2 or 3 hours. Homework which will be discussed in the classroom

Module 2

 

Regular 2 or 3 hours lectures with occasional in-course tests.

Assessment methods

Module 1

Written and oral exams with occasional in-course tests

Module 2

Written and oral exams.

 

Teaching tools

Module 1

Tablet used as a blackboard

Module 2

 

Supplementary notes may be distributed during the course itself.

Links to further information

http://www.ams.org/mathscinet/MRAuthorID/618511

Office hours

See the website of Daniele Ritelli

See the website of Enrico Bernardi