Program Overview

Program Overview

The course provides a simple yet rigorous introduction to linear differential equations of the second order, including homogeneous and non-homogeneous equations, and their resolution in the case of constant coefficients. The main applications of these equations are also presented, including forced and damped oscillations and the phenomenon of resonance.

Course Description

The course is designed for use in university courses that cover the subject, particularly Mathematical Analysis 2 and Differential Equations. The prerequisites for the course are differential and integral calculus for functions of one real variable.

Course Structure

The course consists of three weeks:

• Week 1: Introduction to linear differential equations of the second order, including the theorem of structure, and resolution of homogeneous equations with constant coefficients.
• Week 2: Non-homogeneous equations, including cases of simple but relevant forcing and the general case.
• Week 3: Physical models, including free, forced, and damped oscillations, and resonance.

Learning Outcomes

By actively participating in this course, students will be able to:

• Understand linear differential equations of the second order
• Apply the theorem of structure to describe the set of solutions of homogeneous equations
• Solve homogeneous equations with constant coefficients
• Interpret the structure of the set of solutions
• Understand non-homogeneous equations
• Solve non-homogeneous equations with simple forcing
• Apply the method of variation of constants to find particular solutions in non-homogeneous equations
• Solve non-homogeneous equations with general forcing
• Interpret the physical meaning of the solutions of non-homogeneous equations
• Understand free, forced, and damped oscillations
• Analyze resonance
• Apply differential equations to physical phenomena

Target Audience

The course is designed for Bachelor of Science students, particularly those in MOOCs (Massive Open Online Courses).

Duration and Level

The course lasts for three weeks and is at the EQF (European Qualifications Framework) level 6.

Subject Area

The subject area of the course is Mathematics, specifically linear differential equations of the second order.

License

The course is distributed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

Instructors

The course instructors are:

• Marco Bramanti, Professor of Mathematical Analysis at the Politecnico di Milano
• Domenico Brunetto, Collaborator in scientific research and teaching at the Politecnico di Milano
• Monica Conti, Professor of Mathematical Analysis at the Politecnico di Milano
• Giulio Magli, Professor of Rational Mechanics at the Politecnico di Milano

Assessment and Certification

The course includes quizzes and a final questionnaire. Students who complete the course and achieve a score of at least 60% on the quizzes will receive an Open Badge certificate. The Open Badge is not an official certificate and does not grant university credits or degrees.