# Math 634: Theory of Ordinary Differential Equations

## Contents

## Catalog Information

### Title

Theory of Ordinary Differential Equations.

### Credit Hours

3

### Prerequisite

Math 334, 341; or equivalents.

### Description

A rigorous treatment of the theory of differential equations.

## Desired Learning Outcomes

This course is aimed at graduate students in Mathematics as well as graduate students in Physics and Engineering. This course contributes to all the expected learning outcomes of the Mathematics M.S. and Ph.D. The topics include the existence, uniqueness, and continuation of solutions to differential equations, linear differential equations, and stability of solutions for differential equations.

### Prerequisites

Students should have taken Math 334 and Math 342 prior to taking this course. Math 342 provides a rigorous background of analysis that is needed to understand many of the proofs in this course. Math 334 provides the students an understanding of basic properties of differential equations and techniques used to solve differential equations.

### Minimal learning outcomes

- Solutions to ordinary differential equations
- Existence of solutions
- Uniqueness of solutions
- Continuation of solutions
- Gronwall’s inequality
- Dependence on parameters
- Contraction mapping principle

- Linear differential equations
- Linear systems with constant coefficients
- Jordan Normal Form
- Fundamental solutions
- Variation of constants formula
- Floquet Theory for periodic solutions

- Stability and instability
- Stability and asymptotic stability
- Lyapunov functions
- Bifurcations

- Poincare-Bendixson Theory
- Invariant sets
- Omega limit sets
- Limit cycles

### Textbooks

Possible textbooks for this course include (but are not limited to):

### Additional topics

These are at the instructor's discretion as time allows examples are: Stability of nonautonomous equations, Fredholm alternative, normal forms, Hamiltonian dynamics, control theory, stable manifold theorem, and center manifold theorem.