Course 2024-2025 a.y.

• Examples of ODEs from real-world applications
• Elementary techniques for solving simple differential equations, such as separable and linear ODEs.
• Linear algebra: eigenvalues, eigenvectors, diagonalization, generalized eigenspaces, Jordan decomposition.
• Linear systems of ODEs with constant coefficients: general results and structure of the space of solutions, matrix exponential.
• Basic notions of topology in the Euclidean space.
• Spaces of continuous functions: uniform convergence, completeness, compactness.
• Nonlinear systems of ODEs - local theory: Banach contraction theorem, existence and uniqueness.
• Linearization of autonomous systems around equilibria and classification of equilibrium points in dimension two.
• Nonlinear systems of ODEs - global theory: properties of maximal solutions, basics of the theory of dynamical systems (Newtonian systems and statements of some cornerstone results).

• Know fundamental notions in linear algebra: eigenvalues, eigenvectors, diagonalization, Jordan decomposition.
• Express basic notions and results about spaces of continuous functions.
• Understand the relevance of the fundamental theorems for ODEs and dynamical systems: existence, uniqueness and stability of solutions, stability/instability of fixed points in a dynamical system.

• Solve linear systems of ODEs.
• Provide qualitative description of the solutions of nonlinear ODEs and dynamical systems.
• Make use of the presented methodological tools in applied sciences.

• Lectures
• Practical Exercises
• Individual works / Assignments
• Collaborative Works / Assignments

Face-to-face lectures are standard blackboard presentations, where nonetheless interaction between students and instructor is strongly encouraged.

Exercise sessions are dedicated to practice with the main theoretical notions and results shown during the lectures, looking at specific examples, applications, and exercises.

Assignments will be provided in the form of problem sheets. They are not graded and students can choose to solve them individually or with their peers.

Students will be evaluated on the basis of written exams, which can be taken in one of the two following ways. 

The exam can be split in two partial exams. Each partial may contain multiple-choice questions and open-answer questions; each partial weighs for one-half of the final mark. Multiple-choice questions mainly aim at evaluating the knowledge of the fundamental mathematical notions and the ability to apply these notions to the solution of simple problems and exercises, while open-answer questions mainly aim at evaluating:

• The ability to articulate the knowledge of mathematical notions in a conceptually and formally correct way, adequately using definitions, theorems and proofs.
• The ability to actively search for deductive ideas that are fit to prove possible links between the properties of mathematical objects.
• The ability to apply mathematical notions to the solution of more complex problems and exercises.

The exam can also be taken as a single general exam, which contains both multiple-choice questions and open-answer questions. The general exam covers the whole syllabus of the course and it can be taken in one of the four general sessions scheduled in the academic year. This option is mainly meant for students who have withdrawn from the two-partials procedure or could not follow it. Each type of question contributes in a specific way to the assessment of the students' acquired knowledge. Multiple-choice questions mainly aim at evaluating the knowledge of the fundamental mathematical notions and the ability to apply these notions to the solution of simple problems and exercises, while open-answer questions mainly aim at evaluating:

• The ability to articulate the knowledge of mathematical notions in a conceptually and formally correct way, adequately using definitions, theorems and proofs.
• The ability to actively search for deductive ideas that are fit to prove possible links between the properties of mathematical objects.
• The ability to apply mathematical notions to the solution of more complex problems and exercises.

We will take care to obtain final grades whose distribution follows the grade distribution that is recommended by Università Bocconi.

• Schaeffer, David G.; Cain, John W.: Ordinary differential equations: basics and beyond. Texts in Applied Mathematics, 65. Springer, New York, 2018 (softcover version of the 2016 edition). xxx+542 pp. ISBN: 978-1-4939-8184-7