ORDINARY DIFFERENTIAL EQUATIONS (MA 4204 / MATH 252)
Fall 2025
Instructor
Mike O'Neil (oneil@cims.nyu.edu)
Office hours
Mon 11:00am-12:00pm, 2 MTC 854
Tue 11:00am-12:00pm, WWH 1119
(or by appointment)
Description
A first course in ordinary differential equations. This course covers methods for solving first-order linear and nonlinear equations, existence and uniqueness of solutions, and analytical methods for finding solutions. We will also study second-order linear equations, general theory and Wronskians, constant coefficient theory and mechanical vibrations, variation of parameters, and series solutions. More advanced topics toward the end of the semester will include systems of linear equations, eigenvector methods, qualitative analysis of nonlinear systems of equations, boundary-value problems, and an introduction to Fourier Series and Sturm-Liouville theory. Time permitting, we will also discuss the Laplace Transform and how it can be used to solve ODEs, as well as Green’s function methods for solving differential equations. View the syllabus for more detailed information.
Lecture
Monday & Wednesday 9:30am - 10:50am, Jacobs Hall 202
Recitation
Ashley Sobolewski (as21142@nyu.edu)
Friday 9:30am - 10:50am, 2 MTC 817
Office hours: 4:30pm - 5:30pm, 2 MTC 8th floor
Mike O'Neil (oneil@cims.nyu.edu)
Office hours
Mon 11:00am-12:00pm, 2 MTC 854
Tue 11:00am-12:00pm, WWH 1119
(or by appointment)
Description
A first course in ordinary differential equations. This course covers methods for solving first-order linear and nonlinear equations, existence and uniqueness of solutions, and analytical methods for finding solutions. We will also study second-order linear equations, general theory and Wronskians, constant coefficient theory and mechanical vibrations, variation of parameters, and series solutions. More advanced topics toward the end of the semester will include systems of linear equations, eigenvector methods, qualitative analysis of nonlinear systems of equations, boundary-value problems, and an introduction to Fourier Series and Sturm-Liouville theory. Time permitting, we will also discuss the Laplace Transform and how it can be used to solve ODEs, as well as Green’s function methods for solving differential equations. View the syllabus for more detailed information.
Lecture
Monday & Wednesday 9:30am - 10:50am, Jacobs Hall 202
Recitation
Ashley Sobolewski (as21142@nyu.edu)
Friday 9:30am - 10:50am, 2 MTC 817
Office hours: 4:30pm - 5:30pm, 2 MTC 8th floor
Materials
The following textbook will be used for the course:
- Martin Braun, Differential Equations and Their Applications, 4th ed., Springer, 1993
The following books might be useful as additional references:
- Boyce, DiPrima, and Meade, Elementary Differential Equations and Boundary Value Problems, 11th ed., Wiley, 2017
- Coddington, An Introduction to Ordinary Differential Equations, Dover, 1989
Grading
The overall course grade will be determined from a final numerical weighted average. The following breakdown will be used to compute an overall numerical grade:
- 10% Homework (weekly, lowest grade dropped)
- 25% Preliminary Exam 1
- 25% Preliminary Exam 2
- 40% Final exam (cumulative)
Announcements
The following textbook will be used for the course:
- Martin Braun, Differential Equations and Their Applications, 4th ed., Springer, 1993
The following books might be useful as additional references:
- Boyce, DiPrima, and Meade, Elementary Differential Equations and Boundary Value Problems, 11th ed., Wiley, 2017
- Coddington, An Introduction to Ordinary Differential Equations, Dover, 1989
Grading
The overall course grade will be determined from a final numerical weighted average. The following breakdown will be used to compute an overall numerical grade:
- 10% Homework (weekly, lowest grade dropped)
- 25% Preliminary Exam 1
- 25% Preliminary Exam 2
- 40% Final exam (cumulative)
Announcements
Schedule
Below is an updated list of discussion topics along with any documents that were distributed, homework assigned, etc.| Date | Topics | Materials |
|---|---|---|
| Wed Sep 3 | Overview, 1st order equations | Braun 1.1-1.2 |
| Mon Sep 8 | Integrating factors, Carbon dating | Braun 1.1-1.3 |
| Wed Sep 10 | Separable equations, intervals of existence, orthogonal trajectories | Braun 1.4, 1.8(c) |
| Mon Sep 15 | Exact equations | Braun 1.9 |
| Wed Sep 17 | Existence and uniqueness, Picard iterations | Braun 1.10 |
| Mon Sep 22 | 2nd-order linear differential equations | Braun 2.1 |
| Wed Sep 24 | Constant coefficient equations; reduction of order | Braun 2.2 |
| Mon Sep 29 | Inhomogeneous equations, variation of parameters | Braun 2.3-2.4 |
| Wed Oct 1 | Series solutions | Braun 2.8 |
| Mon Oct 6 | Prelim Exam 1 | |
| Wed Oct 8 | Series solutions: Singular points, regular singular points | Braun 2.8.1 |
| Tue Oct 14 | Series solutions: Method of Frobenius | Braun 2.8.2 |
| Wed Oct 15 | The Laplace Transform | Braun 2.9 |
| Mon Oct 20 | The Laplace Transform: Solving ODEs | Braun 2.9-2.11 |
| Wed Oct 22 | The Laplace Transform: Delta functions, convolutions | Braun 2.12-2.13 |
| Mon Oct 27 |
The Laplace Transform: Convolutions Systems of differential equations |
Braun 2.13 Braun 3.1-3.4 |
| Wed Oct 29 | Systems of diff eqns | Braun 3.8-3.9 |
| Mon Nov 3 | Systems of diff eqns: Eigenvector/value methods, matrix exponential | Braun 3.8-3.10 |
| Wed Nov 5 | Qualitative theory: Stability, linearization | Braun 4.1-4.3 |
| Mon Nov 10 | Prelim Exam 2 | |
| Wed Nov 12 | Qualitative theory: Stability, phase plane | Braun 4.1-4.3 |
| Mon Nov 17 | Qualitative theory: Phase portraits, bifurcations | Braun 4.4, 4.7, 4.9 |
| Wed Nov 19 |
2-point boundary value problems PDEs: Separation of variables |
Braun 5.1-5.3 |
| Mon Nov 24 | ||
| Wed Nov 26 | NO CLASS -- FRIDAY SCHEDULE | |
| Mon Dec 1 | ||
| Wed Dec 3 | ||
| Mon Dec 8 | ||
| Wed Dec 10 | ||
| Mon Dec 15 | Final exam 8:00am - 10:00am |