School of Arts and Sciences
MA205: Differential Equations
Course Syllabus
Mission Statement: “Daniel Webster College educates purposeful men and women for entry,
advancement and advanced studies in professional fields through programs which emphasize the
integration of theory and practice through interactive teaching and learning in professional and liberal
studies.”
Course Number/Section:
Course Title:
Semester/Term Dates:
Class Meeting Time:
Class Location:
Prerequisites:
Co-requisites:
Credits:
MA205-A
Differential Equations
Spring 2014
Monday, Wednesday, Friday, 11:00 – 12:10
DWH 226
MA202
None
4.0
Instructor Contact Information
Instructor:
Office Location:
Office Hours:
Email:
Phone:
Timothy D. Kostar
DWH 109 G
Monday, Wednesday, Friday, 9:00 – 11:00
kostar_timothy@dwc.edu
603-577-6064
Course Description
This is the fourth course in a standard engineering/physical science mathematics sequence. It provides
the background necessary for many engineering courses. Our main goals are to acquire a good
understanding of the concepts in ordinary differential equations, to apply the appropriate techniques for
their solution, and to examine various applications involving differential equations.
The main part of this course is focused on obtaining analytical solutions for ordinary differential
equations. The theory of differential equations is presented to the extent that it provides information
about, and is useful to, the solution of differential equations. Most differential equations of interest to
engineers come about through mathematical modeling of physical systems. Therefore, students will be
introduced to the concepts of mathematical modeling and, in particular, differential modeling. This
course prepares students for many engineering courses including Linear Algebra & Intro to Numerical
Methods, Electrical Engineering, Heat Transfer, Vibrations, and Control Systems Analysis.
Course Textbook Information
Erwin Kreyszig, “Advanced Engineering Mathematics”, 10th ed, John Wiley and Sons, Inc., New York
(2011), ISBN 978-0-470-45836-5
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Course Objective
The objective of this course is to have the student develop a good understanding of the characteristics of
differential equations and the use of these characteristics to categorize them for solution, obtain
analytical solutions to common 1st and 2nd order differential equations, and apply differential modeling
techniques to develop differential equations describing simple physical phenomena.
Course Outcomes (measurements in parentheses)
The student will demonstrate ability to
1) Identify and solve a variety of first order, ordinary differential equations (HW 2, 3, 4, Exam #1).
2) Obtain the solution to second-order, constant coefficients, homogeneous differential equations
(HW 5, Exam #2, Exam #3, Final Exam).
3) Obtain the solution to second-order, constant coefficients, non-homogeneous differential
equations by the methods of Undetermined Coefficients and Variation of Parameters (HW 6, 7, 8,
Exam #2, Exam #3, Final Exam).
4) Obtain the solution to the Cauchy-Euler equation (HW 9, Exam #3).
5) Obtain a Power Series solution for a second-order, non-constant coefficients, homogeneous
differential equation (HW 10, 11, Exam #4, Final Exam).
6) Use Laplace transforms to solve ordinary differential equations with constant coefficients (HW 11,
12, Exam #4, Final Exam).
7) Use mathematical modeling to derive, and appropriate methods to solve, differential equations
governing basic systems (Exam #1, Exam #3, Final Exam, Modeling Project).
General Education Competencies
The General Education category that this course supports is critical thinking. Specifically, the following
competencies are addressed:
-
Demonstrate the ability to describe and evaluate the logic of an argument or analysis.
Apply quantitative reasoning skills to solve problems.
Students with Disabilities
Daniel Webster College is committed to compliance with Section 504 of the Rehabilitation Act of 1973 and
its regulations. The school does not discriminate on the basis of disability in admission or access to, or
treatment or employment in, its programs and activities.
In accordance with the Americans with Disabilities Act, any student who has a documented physical,
learning, or emotional disability* will be provided with reasonable accommodations designed to meet his
or her needs. Before any such assistance can occur, it is the responsibility of the student to see that
documentation is on file with the ADA Coordinator and that a Reasonable Accommodation Plan has been
developed. Once this is in place the student may request a copy of the plan go to all or some of his/her
instructors so that they may provide the agreed upon accommodations. Students with a disability may
request an accommodation by contacting ADA Coordinator Dr. Kathy Hipp, Associate Dean of Arts and
Sciences, at 603-577-6659 or hipp@dwc.edu.
*Documentation cannot be more than three years old.
Academic Honesty
Intellectual curiosity is at the heart of the academic enterprise. Students, faculty and administration at
Daniel Webster College consider such violations as cheating and plagiarism to be so unethical as to call
into question whether the violator should continue as a member of the College community.
Transcripts that misrepresent academic performance not only endanger students’ chances for success in
their careers but also damage the integrity and reputation of the institution.
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Student Honor Pledge
Daniel Webster College believes that all students have the right to learn in an academic community that
insures fair competition, and respects truth and honesty. Academic dishonesty is not tolerated at Daniel
Webster College. The Student Honor Pledge is intended to create a community of fairness, respect and
responsibility in the pursuit of academic enterprise. All students are expected to abide by the Student
Honor Pledge.
I pledge on my honor, as a student at Daniel Webster College, that I have neither given nor
received any unauthorized aid on this assignment/examination.
For more information regarding Daniel Webster College’s ethical standards, please refer to the current
college catalog.
Grading Scale
The following scale is used to assign letter grades:
A = 93+
C+= 76-79
A- = 90-92
C = 70-75
B+= 87-89
D = 65-69
B = 83-86
F = Below 65
B- = 80-82
Expectations
Homework problems from the text will be assigned on a regular basis and are subject to change. A
limited number of other assignments not in the text might be given as supplemental exercises. It is
expected that the interested student will endeavor to solve these problems. Remember that you
cannot adequately learn this material without practicing the methods. Talk to me in advance if
you have reasonable reason(s) for not being able to turn in your homework on time. To be fair to all
students, late homework or other assignments are given a 33% reduction in points per day that the
homework / assignment is late. A “day” is defined as 24 hours, beginning at the start of class time.
In addition, there will be a Modeling Project involving the independent mathematical modeling of a
dynamic system, obtainment of the solution, and study of the system response to parametric changes.
Finally, in support of the student evaluation process, there will be four (4) in-term Exams, and one
(1) comprehensive Final Exam. In computing your average for the four in-term exams, the lowest
test score will be dropped. Under no circumstances will makeup exams be given. If you miss a test for
any reason (such as illness, death or marriage of a family member or friend, you’re attending a rock
concert in Boston) you will receive a grade of zero for that test.
Active class participation is expected. In this course you will be expected to act in a professional
manner. Among other things, this includes showing up on time prepared for the task at hand. This shall
include not just being on time for class, but also for any and all additional outside meetings you will have
with group work. You will be expected to read assigned chapters/tutorials before coming to class and be
ready to actively participate. Classroom activities such as chatting, use of computer, ipod, cell phone and
other electronics are not allowed.
Students may ask questions of one another when working on out-of-class assignments. However, each
student/team must do their own work. A first occurrence of academic dishonesty will result in a
zero for that assignment for all who are involved. A second occurrence will result in an F for the course.
Note that all such occurrences must be reported in writing to the Chief Academic Officer who may
prescribe additional penalties.
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Course Evaluation
Homework
15%
Modeling Project
5%
Average of 3 of 4 Exams
50%
Final Exam
30%
________________________________________
Total
100%
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Planned Schedule of Activities
Schedule may be modified, with announcements posted on-line.
Week
1
2
3
4
5
6
7
8
9
10
01/13
01/17
Classes
start
Tues.
01/20
01/24
Reading
Chap. 1
Chap. 1
Chap. 1
02/03
02/07
Chap. 1
02/10
02/14
Chap. 2
03/03
03/07
03/10
03/14
03/17
03/21
Chap. 2
Chap. 2
Chap. 2
Chap. 2
03/24
03/28
Chap. 5
12
03/31
04/04
Chap. 5
13
04/07
04/11
04/14
04/18
15
04/21
04/25
16
04/28
04/29
Mon
and
Tues
only
04/30
05/06
Starts
on Wed.
16
Differential Equations and their solution.
1st Order ODE’s, Exact differential equations.
Separable differential equations. Integrating Factors. Bernoulli eqn.,
Applications and modeling.
1st Order ODE’s, Exact differential equations.
Separable differential equations. Integrating Factors. Bernoulli eqn.,
Applications and modeling.
1st Order ODE’s, Exact differential equations.
Separable differential equations. Integrating Factors. Bernoulli eqn.,
Applications and modeling.
2nd Order ODE’s, Homogeneous, constant coefficients. Non-homogeneous
,constant coefficient. The method of undetermined coefficients. Variation
of parameters. The Cauchy-Euler equation. Applications and modeling.
Exam #1: Friday Chap. 1
2nd Order ODE’s, Homogeneous, constant coefficients. Non-homogeneous
,constant coefficient. The method of undetermined coefficients. Variation
of parameters. The Cauchy-Euler equation. Applications and modeling.
HW #2
HW #3
HW #4
HW #5
HW #6
11
14
Assignments
HW #1
01/27
01/31
02/17
02/21
No Mon.
Tues is
Mon.
sched.
02/24
02/28
Topics
Chap. 6
Chap. 6
Chap. 6
2nd Order ODE’s, Homogeneous, constant coefficients. Non-homogeneous
,constant coefficient. The method of undetermined coefficients. Variation
of parameters. The Cauchy-Euler equation. Applications and modeling.
2nd Order ODE’s, Homogeneous, constant coefficients. Non-homogeneous
,constant coefficient. The method of undetermined coefficients. Variation
of parameters. The Cauchy-Euler equation. Applications and modeling.
Exam #2: Friday Chap. 2 (partial)
Spring Break
2nd Order ODE’s, Homogeneous, constant coefficients. Non-homogeneous
,constant coefficient. The method of undetermined coefficients. Variation
of parameters. The Cauchy-Euler equation. Applications and modeling.
Power series solutions (ordinary point).
Power series solutions (singular point).
Bessel’s equation and Bessel functions.
Exam #3: Friday Chap. 2 (partial) and Chap. 5 (partial)
Definition, Existence, and basic properties of the Laplace transform.
The inverse transform and the convolution.
LT solution of linear differential equations with constant coefficients.
HW #7
HW #8
HW #9
Modeling Project assigned
HW #10
HW #11
HW #12
HW #13
LT solution of linear differential equations with discontinuous homogeneous
terms.
Exam #4: Friday Chap. 5 (partial) and Chap. 6
Review for Final Exam
Modeling Project due
Final Exams
Week
Comprehensive Final Exam
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Relationship of Course to Program Outcomes
Mechanical Engineering
(slight, moderate, substantial)
Outcome
Level of
contribution
Outcome
Level of
contribution
Outcome
a
b
c
d
e
f
g
h
i
j
k
l
m
a
b
c
d
substantial
h
e
f
slight
i
moderate
j
k
g
moderate
l
m
slight
slight
Description of Outcome
an ability to apply knowledge of mathematics, science, and engineering
an ability to design and conduct experiments, as well as to analyze and interpret data
an ability to design a system, component, or process to meet desired needs within realistic
constraints …
an ability to function on multi-disciplinary teams
an ability to identify, formulate, and solve engineering problems
an understanding of professional and ethical responsibility
an ability to communicate effectively
the broad education necessary to understand the impact of engineering solutions in a
global, etc., societal context
a recognition of the need for, and the ability to engage in life-long learning
a knowledge of contemporary issues
an ability to use the techniques, skills, and modern engineering tools needed for
engineering practice
an ability to apply principles of engineering, basic science, and mathematics (including
multivariate calculus and differential equations) to model, analyze, design, and realize
physical systems, components or processes.
an ability to work professionally in both thermal and mechanical systems areas.
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Relationship of Course to Program Outcomes
Aeronautical Engineering
(slight, moderate, substantial)
Outcome
Level of
contribution
Outcome
Level of
contribution
Outcome
a
b
c
d
e
f
g
h
i
j
k
l
m
n
a
b
c
d
substantial
h
e
f
slight
i
j
k
l
g
moderate
m
n
moderate
Description of Outcome
an ability to apply knowledge of mathematics, science, and engineering
an ability to design and conduct experiments, as well as to analyze and interpret data
an ability to design a system, component, or process to meet desired needs within realistic
constraints …
an ability to function on multi-disciplinary teams
an ability to identify, formulate, and solve engineering problems
an understanding of professional and ethical responsibility
an ability to communicate effectively
the broad education necessary to understand the impact of engineering solutions in a
global, etc., societal context
a recognition of the need for, and the ability to engage in life-long learning
a knowledge of contemporary issues
an ability to use the techniques, skills, and modern engineering tools needed for
engineering practice
a knowledge of aerodynamics, aerospace materials, structures, propulsion, flight
mechanics, and stability and control
design competence that includes integration of aeronautical topics
an ability to develop flight test plans and conduct in-flight experiments, as well as to
analyze, etc., the resulting data
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Modeling Project Assessment Rubric
The primary learning outcomes of this project include:
-
Demonstrate a general understanding of the topic including background, motivation for doing the
work, the relevance of the work, specific objectives.
Demonstrate the ability to apply fundamental principles to mathematically model a physical
system.
Demonstrate the ability to apply standard analytical methods to obtain a solution to the governing
differential equation.
Demonstrate the ability to effectively and systematically analyze the response of the modeled
system based on the parameters of the problem.
Demonstrate advanced technical writing skills including completeness and clarity of presentation,
grammar.
Assessment:
Performance
Indicator
Criteria
Value
Beginning (1)
Developing (2)
Accomplished (3)
Exemplary (4)
Topic
Knowledge
Little to no
understanding of
topic details.
Showed some
understanding of
topic details.
Demonstrated an
understanding of the
topic details.
Demonstrated
understanding of topic
and related topic details.
Mathematical
Modeling
Did not
demonstrate
ability to perform
mathematical
modeling.
Demonstrated
some ability to
perform
mathematical
modeling.
Demonstrated ability to
perform mathematical
modeling.
Demonstrated ability to
perform mathematical
modeling and showed
deep understanding of
fundamental principles.
Mathematical
Solution
Did not
demonstrate
ability to obtain
mathematical
solution.
Demonstrated
some ability to
obtain
mathematical
solution.
Demonstrated ability to
obtain mathematical
solution.
Demonstrated ability to
obtain mathematical
solution and showed
deep understanding of
mathematical principles.
System
Analysis
Did not show
ability to perform
basic system
analysis.
Showed some
ability to perform
basic system
analysis.
Demonstrated ability to
perform basic system
analysis.
Demonstrated ability to
perform advanced
system analysis showing
deep understanding of
the models relationship
to the physical system.
Technical
Writing
Did not
demonstrate
advanced
technical writing
skills.
Demonstrated
some technical
writing skills.
Demonstrated ability to
write a clear and
comprehensive
technical report.
Demonstrated ability to
write a clear and
comprehensive technical
report, including
grammatical details.
Total
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