Department of Mathematics and Physics
§ I: MATH 3309 Advanced Dierential Equations Syllabus
Catalog Description
Prerequisite: MATH 2204 or MATH 2205. The topics covered: existence and uniqueness theorem, Picard
method of solution of systems of dierential equations, classes of non-linear ordinary dierential equations
and solution methods, systems of dierential equations, qualitative analysis of systems of ordinary dier-
ential equations, classication of equilibrium points, linearization and eigenvalues method, and Lyapunov
method. 3 credits
Required Textbook
Dierential Equations and Their Applications, by Martin Braun, Springer, 4e, ISBN 9781111827052 (1993).
All students, regardless of their instructors policies regarding online homework, are encouraged to be-
come familiar with and use online homework as a tool to augment their study of dierential equations.
Course Objectives
The course, MATH 3309, provides a continuation of the theory and solution of ordinary dierential equations
(ODEs). The emphasis is on developing the mathematical properties and solution methods that characterize
ordinary dierential equations, including a more detailed examination of complex eigenvalues and eigenvec-
tors, stability, the phase plane. In addition,
1. Working with systems of ordinary dierential equations and non-linear ordinary dierential equations
is also stressed.
2. Developing and understanding and appreciation of the qualitative behavior of the solution vis-a-vis
orbits and phase plane portraits;
3. Being able to formulate and nd solutions to more complex mathematical problems encountered in the
applied sciences and engineering involving dierential equations.
The emphasis is on improved critical thinking skills with regard to using extending the methods learned in
MATH 2204/MATH 2205 to solve more dicult problems involving dierential equations. Theory and analysis
is stressed throughout, however the course also requires that the student develop prociency in working with
solution methods for ODEs that are covered in the text.
Student Learning Outcomes
After successfully completing this course the expectation is that students will be able to:
1. Apply the Picard method to the proof of the existence and uniqueness theorem.
2. Identify classes of non-linear ordinary dierential equations.
3. Apply an appropriate method for the solution of non-linear ordinary dierential equations.
4. Analyze qualitative properties of systems of ordinary dierential equations.
Required Curriculum Content
Key topics covered include:
1. First Order Dierential Equations (ODEs):
Initial value problems and the solution of rst order ODEs, including separable equations, and an analy-
sis of exact solutions.
2. Existence and uniqueness theorems, Picard iteration.
3. Second order linear ODEs:
Algebraic properties of solutions, constant coecient ODEs, nonhomogenous equations, and variation
of parameter. Mechanical vibrations. Series solutions, Laplace transforms, and solving problems involv-
ing discontinuous right-hand-sides.
4. Systems of ODEs:
Algebraic properties of linear systems, including their solution; and properties of linear transformations,
eigenvalues and eigenvectors, complex roots, and nonhomogenous equations.
5. Solution methods:
Series solutions (singular points, regular singular points), the method of Frobenius, and Laplace trans-
6. Linear systems of ODEs:
Stability and solution of linear systems and multiple eigenvalues, nonhomogenous systems and varia-
tion of parameter methods. Orbits and phase portraits.
7. Non-linear systems of ODEs:
Stability, equilibrium points, and linearization. Phase portraits of non-linear systems, Poincare-Bendixson
Theorem, bifurcation of solutions.
All sections of MATH 3309 Advanced Dierential Equations will cover, as a minimum, the material from Dier-
ential Equations and Their Applications, by Martin Braun, Springer, 4e, ISBN 9781111827052 (1993), as listed:
Sec Textbook Topic
Chapter 1 First-order dierential equations
1.2 First-order linear dierential equations
1.4 Separable equations
1.9 Exact equations
1.10 The existence-uniqueness theorem; Picard iteration
Chapter 2 Second-order linear dierential equations
2.1 Algebraic properties of solutions
2.2 Linear equations with constant coecients
2.3 The nonhomogenous equation
2.4 The method of variation of parameters
2.6 Mechanical vibrations
2.8 Series solutions
2.10 Some useful properties of Laplace transforms
2.11 Dierential equations with discontinuous right-hand sides
Chapter 3 Systems of dierential equations
3.1 Algebraic properties of solutions of linear systems
3.6 Solution of simultaneous linear equations
3.7 Linear transformations
3.8 The eigenvalue-eigenvector method of nding solutions
3.9 Complex roots
3.10 Equal roots
3.12 The nonhomogenous equation
Chapter 4 Qualitative theory of dierential equations
4.1 Introduction
4.2 Stability of linear systems
4.3 Stability of equilibrium solutions
4.4 The phase-plane
4.6 Qualitative properties of orbits
4.7 Phase portraits of linear systems
Department Syllabus for MATH 3309, Spring 2019 Page 2 of 7 Rev. 1.0, January 28, 2019
Sec Textbook Topic
4.9 Introduction to bifurcation theory
Common Department Requirements for MATH 3309
While students in each section of MATH 3309 are assessed by the course instructor, there are general guide-
lines that apply to all sections of MATH 3309. These include:
Calculators and other electronic devices are not allowed on any exams.
Department Syllabus for MATH 3309, Spring 2019 Page 3 of 7 Rev. 1.0, January 28, 2019
Department, College and University Expectations and Policies
It is important that students familiarize themselves with a range of policies and guidelines that have been es-
tablished by the Department of Mathematics and Physics, the College of Arts and Sciences, and the University
of New Haven. These are an integral part of the syllabus for this course.
Adding/Dropping a Class
The nal day to drop this course without it appearing on your transcript is discussed on the
Academic Schedules and Registration web page. After the rst week of class, self-service registration will
not be enabled for students to directly add or drop classes. Students should contact the Registrars oce
directly or the Academic Success Center for assistance with adding and dropping courses during this time.
Attendance Regulations
University attendance policy guidelines require that:
Students are expected to attend regularly and promptly all their classes, appointments, and exercises. While the
university recognizes that some absences may occasionally be necessary, these should be held to a minimum.
A maximum of two weeks of absences will be permitted for illness and emergencies. The instructor has the
right to dismiss from class any student who has been absent more than the maximum allowed. A dismissed
student will receive a withdrawal (W) from the course if they are still eligible for a withdrawal per the university
Withdrawal from a Course policy, or a failure (F), if not. A student who is not ocially registered in the course
is not permitted to attend classes or take part in any other course activities. Students absent from any class
meeting are responsible for making up missed assignments and examinations at the discretion of the instructor.
Students are to adhere to the policy attendance policy guidelines outlined in the University Catalog under the
heading, Attendance Regulations, found online in the Undergradaduate Catalog or alternatively found in the
Student Handbook on pp.4849.
Religious Observance Policy for Students
The University of New Haven respects the right of its students to observe religious holidays that may neces-
sitate their absence from class or from other required university-sponsored activities. Students who wish
to observe such holidays should not be penalized for their absence, although in academic courses they are
responsible for making up missed work. The College provides that,
Instructors should try to avoid scheduling exams or quizzes on religious holidays, but where such conicts occur
should provide reasonable accommodations for missed assignment deadlines or exams. If a class, an assign-
ment due date, or exam interferes with the observance of such a religious holiday, it is the students responsibil-
ity to notify their instructor, preferably at the beginning of the term, but otherwise at least two weeks before the
More information about religious observance policies can be found in the Student Handbook on pp.4849
under the heading, Attendance Policies: Religious Observance Policy for Students.
Withdrawal from a Course
Students wishing to withdraw must submit a request for an ocial course withdrawal in writing using the on-
line Course Withdrawal Form, or alternatively complete and hand in the pdf based Course Withdrawal Form.
The nal date to request a withdrawal is listed in the Academic Calendar. This request must be submitted to
the Registrars Oce and signed by the International Oce if you are an international student. The grade of
W will be recorded, but the course will not aect the GPA.
Incomplete Grade Policy
A grade of Incomplete (INC) is given only in special circumstances and indicates that the student has been
given permission by the instructor to complete required course work (with the same instructor) after the end
Please note that it is the responsibility of the student to assure that the required paperwork and documentation is completed by the deadline.
Department Syllabus for MATH 3309, Spring 2019 Page 4 of 7 Rev. 1.0, January 28, 2019
of the term. In the absence of the instructor a student should contact the Department Chair. Students need
to examine carefully the changed guidelines pertaining to INC grades, specically:
To remove the INC grade, the student must complete all required course work in timely fashion as stipulated by
the instructor but no later than the end of the following term. Fall and intersession course incomplete grades
must be completed no later than the last day of the spring term. Spring and summer course incomplete grades
must be completed no later than the last day of the fall term.
If the course work is not submitted within the allotted time, the INC grade will be changed to an F shortly after
the deadline by the Oce of the University Registrar. Students will be notied via campus email at least two
weeks prior to the change of grade process.
The University policy on incomplete grades is discussed in the Academic Catalog under the heading, Incom-
plete (INC) Grade Policy.
Academic Integrity Policy and Procedures
The University of New Haven expects its students to maintain the highest standards of academic conduct.
Academic dishonesty is not tolerated at the University. To know what it is expected, students are responsible
for reading and understanding the statement regarding academic honesty in the Student Handbook. Specif-
ically, students are required to adhere to the Academic Integrity Policies specied in the Student Handbook,
i.e., on pp.6673.
Please ask your instructor about their expectations regarding permissible or encouraged forms of student
collaboration if there is any confusion about this topic. The Department of Mathematics and Physics fully
adheres to the Academic Integrity Policy:
Academic integrity is a core university value that ensures respect for the academic reputation of the University,
its students, faculty and sta, and the degrees it confers. The University expects that students will conduct
themselves in an honest and ethical manner and respect the intellectual work of others. Please be familiar with
the Universitys policy on Academic Integrity. Please ask about expectations regarding permissible or encouraged
forms of student collaboration if they are unclear.
Coursework Expectations
This course will require signicant in-class and out-of-class commitment from each student. The University
estimates that a student should expect to spend two hours outside of class for each hour they are in a
class. For example, a three credit course would average six [6] hours of additional work outside of class.
Coursework expectations are detailed in the Academic Catalog under the heading, Course Work Expectations.
Please note, that MATH 3309 is a 3-credit course, and as such requires a total of 9 hours per week invested
in study and homework for the average student.
Commitment to Positive Learning Environment
The University adheres to the philosophy that all community members should enjoy an environment free of
any form of harassment, sexual misconduct, discrimination, or intimate partner violence. If you have been
the victim of sexual misconduct we encourage you to report this. If you report this to a faculty/sta member,
they must notify our colleges Title IX coordinator about the basic facts of the incident (you may choose to
request condentiality from the University). If you encounter sexual harassment, sexual misconduct, sexual
assault, or discrimination based on race, color, religion, age, national origin, ancestry, sex, sexual orientation,
gender identity, or disability please contact the Title IX Coordinator, Caroline Koziatek at (203)-932-7479 or Further online information about is available at Title IX.
Reporting Bias Incidents
At the University of New Haven, there is an expectation that all community members are committed to cre-
ating and supporting a climate which promotes civility, mutual respect, and open-mindedness. There also
exists an understanding that with the freedom of expression comes the responsibility to support community
Please note that study guidelines are important, i.e., there is substantial evidence that shows that the pass rates for students in math courses decrease
dramatically as the time spent on outside study falls below 2 hours of homework per credit per week.
Department Syllabus for MATH 3309, Spring 2019 Page 5 of 7 Rev. 1.0, January 28, 2019
members right to live and work in an environment free from harassment and fear. It is expected that all mem-
bers of the University community will engage in anti-bias behavior and refrain from actions that intimidate,
humiliate, or demean persons or groups or that undermine their security or self-esteem.
If you have witnessed or are the target of a bias-motivated incident, please contact the Oce of the Dean
of Students at 203-932-7432 or Campus Police at 203-932-7014. Further information about this and other
reporting options may be found at Report It.
University Support Services
The University recognizes students often can use some help outside of class and oers academic assistance
through several oces. In addition to discussing any academic issues you may have with your instructor,
advisor, or with the the courses or department coordinator or chair, the University provides these additional
resources for students:
The Center for Academic Success and Advising (CASA)
The Academic Success Center is located in Maxcy 208 for help with your academic studies, or call 203-932-
7234 to set up an appointment.
University Writing Center
The mission of the Writing Center (an expansion of the Writer to Writer peer-tutoring program) is to provide
high-quality tutoring to undergraduate and graduate students as they write for a wide range of purposes and
audiences. Tutors are undergraduate and graduate students and they work with students at any stage in the
writing process; Bring in your assignment, your ideas, and any writing done so far. To make an appointment,
register for an account at
The Math Zone
Please contact the Math Zone if you wish to challenge your Math Placement by taking a Math Challenge Exam
or by taking a Math Post Placement Exam. These are discussed more extensively at http://math.newhaven.
edu/mathphysics/placement_html. The Math Zone also provides a range of tutoring and classroom support
service for students taking development math classes.
The Center for Learning Resources (CLR)
The Center for Learning Resources located in Peterson Library, provides academic content support to the
students of the University of New Haven using metacognitive strategies that help students become aware
of and learn to apply optimal learning processes in the pursuit of creating independent learners CLR tutors
focus sessions on discussions of concepts and processes and typically use external examples to help students
grasp and apply the material.
Accessibility Resources Center
Students with disabilities are encouraged to share, in condence, information about needed specic course
accommodations. The Accessibility Resources Center (ARC) provides comprehensive services and sup-
port that serve to promote educational equity and ensure that students are able to participate in the oppor-
tunities available at the University of New Haven. Accommodations cannot be made without written docu-
mentation from the ARC. The ARC is located on the ground oor in the rear of Sheeld Hall. Sheeld Hall
is located in the Residential Quad area, and can be contacted at 203-932-7332. The ADA/Section 504 Com-
pliance Ocer is Rebecca Johnson,, and can be reached by phone at 203-932-7238.
Information on the ARC can be found at
Department Syllabus for MATH 3309, Spring 2019 Page 6 of 7 Rev. 1.0, January 28, 2019
Counseling and Psychological Services
The Counseling Center oers a variety of services aimed at helping students resolve personal diculties and
acquire the balance, skills, and knowledge that will enable them to take full advantage of their experience at
the University of New Haven. Information about the, Counseling and Psychological Services, is available
Department Syllabus for MATH 3309, Spring 2019 Page 7 of 7 Rev. 1.0, January 28, 2019