# Ordinary Differential Equations

MATH-UA 262, *Spring*, 2022

Lectures: Monday and Wednesday 9:30-10:45 pm, Warren Weaver 512

Office Hours: Monday 1:30-20:30 pm and Wednesday 11:00 am - 12:00 noon (Virtual)

# Course Description

Ordinary Differential Equations, ODE for short, was probably my least favorite class in mathematics when I was an undergraduate. It may have something to do with the fact that the class met at 8 am, and that we used a book written by the professor, but at the time I felt that it seemed there were so few equations we could solve (at least out of the universe of equations one could pose), and these solutions often depended on a few tricks, rather than something fundamental. Ugh, I really missed the point!

ODE is where you start to see the power of calculus, the ability to make predictions. Yes, with calculus you can predict the future! Well, more precisely, given the physical laws of the universe (or the markets, or your favorite system), differential equations allow us to forecast how it will evolve into the future. They are indispensable in my field of research, where we work with ODE’s complicated sibling (partial differential equations, which involve multiple dimensions) to predict the response of the weather and climate systems to external forcing, i.e., our greenhouse gas emissions. They are invaluable across the sciences.

ODE also allows us to ask fundamental questions about equations. Do solutions ex- ist (i.e., will this universe end suddenly?) Are solutions unique, or are parallel uni- verses possible next to each other?!? In this, it offers a glimpse at the field of analysis.

The (more modest) goals of this course are to cover these topics:

- methods for solving the (alas) few types of linear first and second order equa- tions that can be solved exactly
- methods for proving existence and uniqueness of solutions
- series solutions for equations with singular points
- systems of linear equations
- nonlinear dynamical systems and phase plane analysis
- boundary value problems
- Green’s functions and Fourier series (if we make it this far).

For more details, please see the course syllabus.

# Textbook

- Braun, Martin,
*Differential Equations and their Applications*, 4th edition, Springer, 1993.