# Ordinary Differential Equations in Real World Situations

Victor Donnay, Professor of Mathematics, Bryn Mawr College

This course for junior and senior math majors uses mathematics, specifically the ordinary differential equations as used in mathematical modeling, to analyze and understand a variety of real-world problems. Among the civic problems explored are specific instances of population growth and over-population, over-use of natural resources leading to extinction of animal populations and the depletion of natural resources, genocide, and the spread of diseases, all taken from current events. While mathematical models are not perfect predictors of what will happen in the real world, they can offer important insights and information about the nature and scope of a problem, and can inform solutions.

The course format is a combination of lecture, seminar and lab. Simulation games, group-work, presentations, and guided inquiry are some of the
pedagogies used in this course, which aims to create a community of learners who have the ability to take what they have learned in one situation and apply it to novel situations, and who can pursue information independently. Beyond the capacity to solve mathematical problems, students are expected to be able to communicate their findings clearly, both verbally and in writing, and to explain the mathematical reasoning behind their conclusions. Learning is assessed through pre- and post-tests and a variety of assignments, including short response papers, quizzes, and a final group project involving an oral report and a 10-15 page paper.

## Course Learning Goals for Instructors and Students

### Instructor Goals

The instructor will teach students to:

• translate (simple) real world situations into ordinary differential equations (the modeling procedure) and
• extract predictive information about the real world situation from the differential equations.
• solve differential equations in a variety of ways: via traditional analytic methods (formulas) as well as by more modern approaches such as
numerical solutions generated by computer programs and by graphical methods that provide qualitative information.
• apply these methods to linear and non-linear equations and systems and see how feedback effects in non-linear systems can lead to unexpected behaviors.

### Student Goals

At the end of the course, a student will:

• Comprehend contemporary applications of computer modeling (e.g. what is meant when a newspaper article reports of new developments in the study of climate change that are predicated by computer modeling).
• Be able to communicate, both in writing an verbally, to explain the mathematical reasoning behind their answer, because solving a mathematical equation is only part of the process of using mathematics.
• Develop their ability to work as independent and self-sufficient learners, with the capacity to learn material on their own, and practice and
proficiency in "What to do when they do not know what to do?"
• Be able to apply what they have learned in one situation to new and different situations (transfer of knowledge).
• Understand mathematical models are not perfect predictors of what will happen in the real world, but they can offer important insights into key elements of a problem
• Be comfortable with not knowing the answer immediately and learning from peers. Students will become part of a community of learners who
support, encourage and learn from one another.