85169 - Differential Equations

Learning outcomes

By the end of the course, the student knows the fundamentals of the theory of ordinary differential equations, that is: existence and uniqueness; equivalence between n-th order scalar equations and a system of n first order equations; the class of first order differential equations solvable in closed form; the change of variable method; the family of linear equations of second order with variable coefficients and their connections with special functions theory; the basics to approximate solutions of differential equations.

Course contents

General Theory.
Preliminary tools: sequences and series of functions: uniform convergence and passage to the limit.

Existence and Uniqueness with the Lipschitz condition. Existence without the Lipschitz condition. Some global properties of solutions. Dependance on data. Nonuniqueness. Approximate solutions.

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

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

Readings/Bibliography

Kenneth B. Howell

Ordinary Differential Equations

An Introduction to the Fundamentals

Second edition 2020

CRC Press

 

 

 

Lecture notes prepared by the instructor.

https://www.dropbox.com/s/lyernxrw6dg3gbl/Main2019_diffeq.pdf?dl=0 

 

Teaching methods

Lessons ex Cathedra. Homework which will be discussed in the classroom

Assessment methods

Written exam with occasional in-course tests. The written examinations detects the capability of the student to solve differential equations giving detailed explanations about the techniques employed in the process. The oral examination is optional: concerns the proofs of theorems exposed during the course. Book and personal notes consulting and calculators are allowed. The use of mobile phones and computers is not allowed.

Teaching tools

Tablet used as a blackboard

Links to further information

https://www.dropbox.com/sh/lm0hsolnlbs1b1q/AAD4IMaIX1XHG7nvpQ6JgX2xa?dl=0

Office hours

See the website of Daniele Ritelli