Math 225 - Introduction to Differential Equations

Spring 2026 - Syllabus - Matthias K. Gobbert

This page can be reached via my homepage at http://www.umbc.edu/~gobbert.

Basic Information

Instructor: Matthias K. Gobbert, gobbert@umbc.edu, office hours Tuesdays and Thursdays 11:30-12:30 in MP 416 in-person, by appointment online, or by e-mail.
Dr. Gobbert has accumulated extensive experience in teaching with state-of-the-art technology. Since participating in the first cohort in the Alternative Delivery Program (ADP) in 2006, he uses hand-writing on tablet laptops for all lectures. These lectures are taped and hosted online for streaming, with the added benefits of allowing for pausing, rewinding, and reviewing. Using the taped lectures for contents delivery, Dr. Gobbert uses a team-based active-learning teaching model, in which students work on problems in learning groups during class. Since 2019, his classes use online comprehension quizzes on the lectures and fully online submission of all assignments, complete with online grading. Since starting online teaching full-time in 2020, the synchronous class meetings are used additionally for student presentations to maximize active student engagement. In 2010, Dr. Gobbert received the University System of Maryland Board of Regents' Faculty Award for Excellence in Mentoring.
This syllabus consists of this frontpage and the sub-pages
- detailed schedule,
- general rules and procedures including grading guidelines, and
- The official ECR website is part of this syllabus.
Classes: Tuesdays and Thursdays, 10:00-11:15, in MP TMP; see the detailed schedule for more information on the coverage, due dates, and timing of the synchronous class meetings.
Note on time commitment: Notice that in a regular semester, you are supposed to spend about 10 to 12 hours per week on this course. If you cannot devote this time, consider if you want to be in this course.
This course will be taught in a flipped classroom format. This means that the contents is delivered asynchronously by taped videos of my lectures that you study before our synchronous class meetings. The synchronous class meetings will use a team-based active-learning teaching model, in which students work on problems in learning groups formed by the instructor with the help of the instructor.
Learning groups: I will form learning groups of three students. These groups are strongly encouraged to also communicate outside of class.
Synchronous class meetings: Our synchronous class meetings are an opportunity for active teamwork with your learning group, while the instructor is available immediately for questions. This and the other strategies above are designed to foster student engagement as well as give you chances to participate more actively and to get to know each other better.
If you have any concerns about any of these items, such as concerns about adequate internet connection, about team work, or special needs related to learning styles, please reach out to me as soon as possible, so I can clarify questions and/or we can work out alternate appropriate approaches and metrics. My goal is definitely that anyone can participate successfully in this course, even if you might need to do it completely asynchronously, but we need to communicate about this.
Required prerequisites: a grade of C or better in Math 152; recommended prerequisites: Math 221 and 251; or instructor approval
This book is highly recommended as reference, but is not required. The intention is to cover the material of the course sufficiently well by the lectures, possibly complemented by specific reading assignments, that I will post online. Stanley J. Farlow, An Introduction to Differential Equations and Their Applications, Dover, 2006. Associated webpage: http://store.doverpublications.com/048644595x.html.
List of errors: http://umbc.edu/~rostamia/farlow-errata.html This list is graciously maintained as benefit to us all by UMBC colleague Rouben Rostamian; if you find additional errors, please inform him at rostamian@umbc.edu.

Grading rule:

Participation	Homeworks	Quizzes	Test 1	Test 2	Test 3	Final Exam
5%	5%	10%	20%	20%	20%	20%

Active participation is the key to success in any class, mathematics or otherwise. This category gives you credit for all associated activities including studying the recorded lectures completely, preparing for the synchronous class meetings as assigned, the active participation in team work, the participation in synchronous class meetings, and adequate professional behavior in all aspects of the course, such as communicating with the instructor, team mates, respectful behavior in communications, and timely submission of work, for instance. The team work in this class is educational in nature, meaning that each and every student has to solve each and every problem; the work is not split up and credit is only given to team members who actively participate (and in particular only if you are present). The purpose of the weight on this category indicates the importance of active partitication in all aspects of the course! To reiterate: You must come to class prepared. This means studying the textbook and watching some of the recordings, so that we have a common starting point and since you cannot learn actively and with others in the class meetings, if you are not prepared. If you have any concerns about this, see the note above.
The online quizzes are administered in the course management system Blackboard, see below, and are due before class. The detailed schedule indicates the planned due dates of these online quizzes, but the Blackboard assignments list the official due dates and times. There will also be in-class quizzes using learning groups formed by the instructor. For instance, they may be designed to initiate class discussion or to give me feedback on your learning. They may be technical or non-technical in nature. An individual Quiz 0 will be given on some material that is critical to your success in this class.
The homework assignments will be posted in the Blackboard site of our course, see below. The detailed schedule indicates the planned due date of each homework, but the Blackboard assignments list the official due dates and times. Working the homework is vital to understanding the course material, and you are expected to work all problems. In the flipped classroom format of this course, we will work on the homework about the associated taped lectures on that topic in class. You should then complete the homework afterwards, and it is then due before the start of the next topic. Homework submission is online as one PDF file in the Assignments area of our Blackboard site. Late submission of homeworks, except Homework 0, cannot be accepted under any circumstances, due to the organizational difficulties associated with the communication with the grader. Homework 0 is required of all students and is accepted late.
The tests and final exam are conventional pencil-and-paper exams. To help you focus on what is relevant, they are closed-book and closed-notes. See the detailed schedule for the dates of the exams.

Additional details or changes will be announced as necessary. See also general rules and procedures for more information. Announcements may be made in class, by e-mail, or in Blackboard. You are responsible for checking your UMBC e-mail address sufficiently frequently.

We will use the course management system Blackboard Ultra for posting of all material (including homework, lesson plans, PDF transcripts, handouts), for links (to tapings of lectures), and for the submission of some work. Notice that Blackboard Ultra is the new version of Blackboard that is more mobile friendly. The main navigation buttons are arranged along the top of the screen, with Content, Announcements, and Gradebook the ones we will use. I will also use Blackboard to send Announcements to the class, which goes to your UMBC account by default. Therefore, you must either check your UMBC e-mail regularly or have the mail forwarded to an account that you check frequently. You should have your notifications in Blackboard turned on! Do not use Blackboard to message me, since I may not find it in a timely fashion; rather use conventional e-mail to my UMBC address listed above.

Course Description

Differential equations are equations, that involve both a function and one or more of its derivatives. The solution to a differential equation is a function of one or more variables. Differential equations arise in an extremely wide array of application areas and are vitally important in the sciences, engineering, and many other fields. This course will stress scalar first-order and second-order ordinary differential equations. We will also cover the solution by Laplace transforms and introduce systems of first-order ordinary differential equations.

This course will develop both a proficiency with the terminology of differential equations and classical analytical solution techniques for ordinary differential equations, with a brief exposure to basic numerical techniques. This approach reflects the fact that it is both the calculation techniques and the fundamental concepts, including the use of numerical methods, that are ubiquitous in the application areas.

Additionally, I will show how to use the software package Matlab to help with some of tasks in this class. Matlab is the -- by far -- most popular mathematical software tool, used in fields ranging from mathematics, statistics, engineering, physical, natural, and life sciences, to economics and business. I use Matlab myself in my professional life, and the idea is to demonstrate how professionals use software tools effectively and appropriately.

Learning Goals

By the end of this course, you should:

understand and remember the key ideas, concepts, definitions, and theorems of the subject. Examples in this course include the classification of differential equations, solvability and uniqueness theorems, and analytical solution techniques. More broadly, you should also understand the purpose of differential equations.
--> This information will be discussed in the lecture. You will apply and use them on homeworks, quizzes, and exams.
be able to apply mathematical theorems and computational algorithms correctly to answer questions, and interpret their results correctly, including potentially non-unique solutions or breakdowns of algorithms. Examples include choosing among several methods to solve a problem and how to react to intermediate solutions found that may indicate a breakdown of the method.
--> The class discussions, homework, and exams address these skills.
appreciate the power of mathematical abstraction and understand how mathematical theory is developed. Classical example of mathematical abstraction in this class are the existence and uniqueness theorem for first-order initial value problems and the theorem that governs the number of fundamental solutions for a linear ordinary differential equation of a given order.
--> These integration goals will be supported by the lectures.
have experience working actively with peers in a group, both on the scale of the class and in a smaller team. Group work requiring communication for effective collaboration with peers and supervisors is a vital professional skill, and the development of professional skills including this networking is a declared learning goal of this course. Additionally, getting to know other students as part of learning groups will prove invaluable for homework and exams.
--> Group discussions and quizzes will contribute to this goal.
have gained experience with a wide variety of teaching and learning techniques, with the goal of encouraging life-long learning.
Examples of teaching techniques include: the use of recorded lectures (that are to be studied before class meetings), the pedagogical technique of a flipped classroom (where we work actively), the user of team-based learning (where you work with peers), and more.
Examples of learning techniques include: learning by reading (e.g., assigned reading before class meetings), learning by hearing (e.g., voice-over in recorded lectures or verbal discussions in class meetings), learning by writing (e.g., answering interpretative questions that require long-hand English answers), learning by talking (e.g., verbal presentations to class), and more.
The reflection on your use of study techniques is a declared learning goal of this course, since it is vital to recognize one's own strengths and weaknesses for best success in live-long learning.
-->All aspects of the course will serve as examples.

Philosophical Underpinning

To provide some context of the more formal learning goals above, I am sharing some deeper thoughts how we fit into the grander scheme of things. The rationale of a state university is to provide a well-educated workforce to the companies in the State of Maryland as well as to the state and local governments themselves. On a fundamental level therefore, you need to able to learn new material as well as have demonstrated evidence of this ability. These are the fundamental purposes of university courses. This requires us to engage in the learning itself and its demonstration; I am trying to say that it is not the solution to a problem that we are after, and in broader thinking it is not even the solution process, but it is your active struggle to learn that we must encourage. This leads me to the following philosophical grade rubric: To earn a passing grade, you need to solve the problem correctly. To earn a good grade, you need to solve the problem correctly and present the solution process completely and professionally. To earn a very good grade, you need to do these and additionally prove that your solution is correct. This rubric comes from the fact that neither you as customer would be willing to accept a faulty solution or unprofessional or incomplete presentation, nor would the companies in the State be served by faulty solutions. These are some of the guiding principles behind my teaching.

Other Information

The recommended literature on my homepage
A brief introduction to Unix/Linux at UMBC
How to get started with Matlab at UMBC (maintained by CIRC)
An introduction to LaTeX gives some pointers how to get started with LaTeX, has a small sample document as well as a larger Template for Project Reports.

UMBC High Performance Computing Facility (HPCF) including general information, list of projects, publications page, and resources for users.
Center for Interdisciplinary Research and Consulting (CIRC)

UMBC Statement of Values for Academic Integrity

Academic integrity is an important value at UMBC. By enrolling in this course, each student assumes the responsibilities of an active participant in UMBC's scholarly community in which everyone's academic work and behavior are held to the highest standards of honesty. Cheating, fabrication, plagiarism, and helping others to commit these acts are all forms of academic dishonesty, and they are wrong. Academic misconduct could result in disciplinary action that may include, but is not limited to, suspension or dismissal. Consult the the UMBC webpage on Academic Integrity at academicconduct.umbc.edu for the UMBC Undergraduate Student Academic Conduct Policy for undergraduate students and the UMBC Graduate School's Policy and Procedures for Student Academic Misconduct for graduate students.