Department of Mathematics and Physics
§ I: MATH 2205 Dierential Equations Syllabus
Catalog Description
A grade of C (not C-) or higher in MATH 1118 or placement by the department. (Note: a student taking MATH
2205 is not eligible to take MATH 2204) Matrices and systems of linear equations, determinants and Cramers
rule, eigenvalues and eigenvectors. First and higher order dierential equations, systems of linear dierential
equations, Laplace transform and its application to the solution of dierential equations. 4 credits.
Required Textbook
Advanced Engineering Mathematics, E. Kreysig, John Wiley, 10e, ISBN 9781111827052 (2011). Students purchas-
ing the text for MATH 2205 should be aware that the text is available both in hard cover (ISBN: 9781118091517)
and electronic version (ISBN: 9780470917336). Both of them come with access code for doing the homework
online with WileyPLUS, which is similar to MyMathLab. Each instructor may choose to use or not use the
online homework package oered by the publisher as part of their assessment, e.g., counting online home-
work as part of the students grade, however once the seal on the software is broken, the package cannot be
returned for a refund.
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 2205, provides a foundation in the theory and solution of ordinary dierential equations
(ODEs) along with an introduction to linear algebra. The emphasis is on connecting mathematical concepts
with practical modeling applications that can be applied to solving problems that arise in engineering and the
applied sciences. Foundational concepts in calculus and linear analysis are stressed throughout the course,
with particular emphasis on the role of linear algebra in setting the framework for understanding linear dif-
ferential equations. The curriculum covers
1. An introduction to applied linear algebra covering content through eigenvalues and eigenvectors;
2. Methods for solving ODEs, in particular the using the Laplace Transform;
3. The structure of the solution to ODEs, and the relationship to underlying mathematical ideas; and,
4. Solutions to mathematical problems in the applied sciences and engineering involving dierential equa-
The emphasis is on improved critical thinking skills with regard to using extending the methods of calculus
to solving elementary 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. Work with matrix-matrix and matrix vector operations, solve systems of equation, nd determinants
of matrices, work with orthogonal matrices, compute eigenvalues and eigenvectors and diagonalize
2. Work with and understand the concept of existence and uniqueness of solutions;
3. Apply principals of linearity to solve a range of problems involving dierential equations;
4. Apply solution techniques to solve rst and second order dierential equation, including homogeneous
and inhomogeneous problems.
5. Apply graphical and numerical methods to solve problems involving ODEs; and,
6. Apply Laplace Transforms to the solution of dierential equations, particular to constant coecient,
linear ODEs.
Required Curriculum Content
Key topics covered include:
1. First Order Dierential Equations (ODEs): Separation of variables; Exactness and integrating factors;
First order linear equations; ODEs reducible to rst order; Homogeneous rs order ODEs.
2. Applications of rst order ODEs: Growth and decay problems; nonlinear models, systems of dierential
3. Second order linear ODEs: Homogeneous equations with constant coecients; Linear dependence and
the Wronskian; Auxiliary equation and solution of homogeneous ODEs; Non-homogeneous ODEs with
constant coecients and the particular solution; The method of undetermined coecients; The method
of variation of parameters.
4. Applications of second order ODEs: Electrical circuits, resonance, coupled systems; Mechanical prob-
5. Series solutions of ODEs: Power series; Singular points of an ODE and their classication; Series solution
about an ordinary point; Series solution about a regular singular point (Frobenius Method).
6. Laplace transforms; Denition and properties of integral transforms; Laplace transforms of simple func-
tions; Laplace transforms of derivatives of functions; Inverse Laplace transforms; Solutions of initial
value problems; Laplace transforms of periodic functions; Laplace transforms of discontinuous func-
tions; and the convolution theorem and its use in computing inverse transforms.
All sections of MATH 2205 Dierential Equations will cover, as a minimum, the material from Advanced Engi-
neering Mathematics, E. Kreysig, John Wiley, 10e, ISBN 9781111827052 (2011), as listed:
Sec Textbook Topic
Chapter 7 Linear Algebra: Matrices, Vectors, Determinants, Linear Systems
7.1 Matrices, Vectors: Addition and Scalar Multiplication
7.2 Matrix Multiplication
7.3 Linear Systems of Equations, Gauss Elimination
7.4 Linear Independence, Rank of a Matrix, Vector Spaces
7.5 Solutions of Linear Systems: Existence, Uniqueness
7.6 For Reference: Second and Third-Order Determinants
7.7 Determinants, Cramers Rule
7.8 Inverse of a Matrix, Gauss-Jordan Elimination
7.9 Vector Spaces, Inner Product Spaces, Linear Transformations
Chapter 8 Linear Algebra: Matrix Eigenvalue Problems
8.1 The Matrix Eigenvalue Problem. Determining Eigenvalues and Eigenvectors
8.2 Some Applications of the Eigenvalue Problem
8.3 Symmetric, Skew-Symmetric and Orthogonal Matrices
8.4 Eigenbases, Diagonalization, Quadratic Forms
Chapter 1 Ordinary Dierential Equations (ODEs)
1.1 Basic Concepts: Modeling
1.2 Geometric Meaning of y
= f (x, y). Direction Fields, Eulers Method
1.3 Separable ODEs: Modeling
1.4 Exact ODEs. Modeling
1.5 Linear ODEs. Bernoulli Equations. Population Dynamics
Chapter 2 Second-Order Linear ODEs
2.1 Homogeneous Linear ODEs of Second Order
Department Syllabus for MATH 2205, Spring 2019 Page 2 of 7 Rev. 1.0, January 28, 2019
Sec Textbook Topic
2.2 Homogeneous Linear ODEs with Constant Coecients
2.3 Dierential Operators
2.4 Modeling of Free Oscillations of a Mass-Spring System
2.5 Euler-Cauchy Equations
2.7 Nonhomogeneous ODEs
2.9 Modeling: Electric Circuits
2.10 Solutions by Variation of Parameters
Chapter 4 Systems of ODEs. Phase Plane. Qualitative Methods
4.0 Basics of Matrices and Vectors
4.1 Systems of ODEs as Models in Engineering Applications
4.2 Basic Theory of Systems of ODEs. Wronskian
4.3 Constant-Coecient Systems. Phase Plane Method
Chapter 6 Laplace Transforms
6.1 Laplace Transform. Linearity. First Shifting Theorem
6.2 Transforms of Derivatives and Integral ODEs
6.3 Unit Step Function (Heaviside Function). Second Shifting Theorem
6.4 Short Impulses. Diracs Delta Function. Partial Fractions
Common Department Requirements for MATH 2205
While students in each section of MATH 2205 are assessed by the course instructor, there are general guide-
lines that apply to all sections of MATH 2205. These include:
Calculators and other electronic devices are not allowed on any exams.
Department Syllabus for MATH 2205, 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 2205, 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 2205 is a 4-credit course, and as such requires a total of 12 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 2205, 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 2205, 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 2205, Spring 2019 Page 7 of 7 Rev. 1.0, January 28, 2019