85169 - Differential Equations

Course Unit Page

Academic Year 2021/2022

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. Solutions by inspection. Change of variables.


Logistic model and its epidemiological applcations: SI (susceptible-infected) model and SIS (susceptible-infected-susceptible ) model. The Kermack–McKendrick SIR (susceptible-infected-recovered) model 


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

Laplace Transform


Suman Kumar Tumuluri

A First Course in Ordinary Differential Equations

ISBN 9780815359838

Book link



D. Ritelli, G. Spaletta:

Introductory Mathematical Analysis for Quantitative Finance

ISBN 978-0-8153-7254-7

Chapters 2, 4, 5 and 6


CRC press 2020 https://www.taylorfrancis.com/books/9781351245111

DOI https://doi.org/10.1201/9781351245111

Complementary Lecture notes prepared by the instructor, available at the link below.


Teaching methods


Lessons ex Cathedra. Homework which will be discussed in the classroom or by Teams. Streaming and recording of the lessons

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.

Evaluation grid for grades

8-23: minimal preparation and ability to analyze, related to an only instrumental mastery of the problems presented in the course, correct use of the methods at elementary level;
24-27: technically adequate preparation but with some limits with respect to the topics covered, good analytical skills, although not particularly articulate, correct use of methods at an intermediate level;
28-30: technically adequate preparation with respect to the topics dealt with, good analytical skills in complex problems, correct use of methods at standard level
30L: excellent and very thorough and exhaustive knowledge of the topics covered in the course, full mastery of the methods also from a theoretical point of view.


In the case of an online exam, the teacher will provide a Dropbox link for downloading the text and one for uploading, where the student will have to upload a pdf file with  the answers given and the solution of the proposed problems.

Teaching tools

Video beamer and tradition blackboard. Computer algebra will be employed to illustrate important topics.

The teaching material presented in class will be made available to the student in electronic format through the university's institutional portal. Username and password are reserved for students enrolled at the University of Bologna.

Office hours can be delivered using Teams

The teacher responds to e-mail messages, duly signed by the student with Name, Surname and matriculation number, and which concern appointment requests or topics that are not covered by the course information presented here.

Links to further information


Office hours

See the website of Daniele Ritelli