Spring 2015
Section 23723
MW 4:30 – 6:00 pm, C 109
Prerequisite: Math 3B & 3E
Math 3F
Differential Equations
Instructor: Michael J. Valdez
Webpage: alameda.peralta.edu/michael-valdez
E-Mail: [email protected]
Phone: (510) 748 – 2137
Office: L 230
Office Hours: MW 3:00 – 4:20 pm
Course Description
Required Materials
Ordinary differential equations: First-order, second-order, and higher-order
equations; separable and exact equations, series solutions, LaPlace
transformations, systems of differential equations.
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Student Learning Outcomes
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Develop problem solving abilities: Synthesize data, translate words
into math language, and construct an abstract model that describes
the problem. (Proof and Deductive Reasoning Skills)
Given data, students will analyze information, and create a graph
that is correctly titled and labeled, appropriately designed, and
accurately emphasizes the most important data content. (Graphing)
Students will be to write and manipulate complex algebraic
expressions and general functions, integrate algebraic and
transcendental functions, and work with sequences and power series
expressions. (Compute, Simplify, and Solve)
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Fundamentals of Differential Equations
by Nagle, Saff, & Snider, 8th Ed
TI-89 Graphing Calculator
General supplies such as pencils, a
large eraser, at least 3 colored pens, a
highlighter, a ruler
Recommended Materials
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Student Solutions Manual for
Fundamentals of Differential Equations
by Nagle, Saff, & Snider, 8th Ed
Schaum’s Outlines for Differential
Equations, 3rd Ed
Lecture Content
1. Differential equations
§ Basic mathematical models, direction fields
§ Solutions of differential equations
§ Classification of differential equations
2. First order differential equations
§ Linear equations
§ Method of integrating factors
§ Separable equations
§ Distinguishing between linear and non-linear equations
§ Method of substitution
§ Exact differential equations
§ Integrating factors, first-order linear equations
§ Numerical approximations, Euler’s method
§ The Existence and Uniqueness Theorem
§ First order difference equations
§ Applications of first order differential equations such as circuits,
mixture problems, population modeling, orthogonal trajectories, and
slope fields
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3. Second order linear differential equations
§ Homogeneous linear equations with constant coefficients
§ Solutions of linear homogeneous equations, the Wronskian
§ Complex roots of the characteristic equation
§ Repeated roots, reduction of order
§ Nonhomogeneous equations, the method of undetermined coefficients
§ Variation of parameters
§ Applications
4. Higher order linear differential equations
§ General theory of nth order linear equations
§ Homogeneous equations with constant coefficients
§ The method of undetermined coefficients
§ The method of variation of parameters
§ Applications of higher order differential equations such as the harmonic
oscillator and circuits
5. Infinite series solutions of second order linear equations
§ Review of power series
§ Series solutions near an ordinary point
§ Euler equations
§ Series solutions near a regular singular point
§ Bessel’s equation
6. The Laplace transform
§ Definition of the Laplace transform
§ Solving initial-value problems
§ Step functions
§ Differential equations with discontinuous forcing functions
§ Impulse functions
§ The convolution integral
7. Systems of first order linear differential equations
§ Matrices
§ Linear algebraic equations, linear independence, eigenvalues,
eigenvectors
§ Basic theory of systems of first order linear differential equations
§ Homogeneous linear systems with constant coefficients
§ Complex eigenvalues
§ Fundamental matrices
§ Repeated eigenvalues
§ Nonhomogeneous linear systems
Student Expectations
Below is a list of expectations I expect from my students:
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Read the textbook as assigned before Monday’s lecture.
Arrive to class on time, especially for exams.
If you need to leave early or arrive late, please do so in manner that does
not disrupt my lecture or your classmates.
Participate during lecture by asking and answering questions.
Complete assignments in a timely fashion that avoids cramming.
Ask homework and project questions during office hours.
Spend at least 1 week preparing for each midterm exam and 1 month
preparing for the final.
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Homework
Grading
I will assign various problems from the text each week and various reading
assignments, which will be due the following Wednesday at 4:30 pm sharp (with
a 5 minute grace period). Your homework will not be accepted if it is submitted
past this time, unstapled, illegible, messy, or not in pencil (colored pens/pencils
allowed for graphs). You are expected to try all of the problems and to check
your work by looking in the back of the book or conversing with a classmate
before turning this in. Remember, no work = no credit. No late homework will
be accepted under any circumstances. That said, I will drop the lowest
homework score from your final grade.
Homework: 20%
Midterm Exams: 30%
Final: 30%
Projects: 10%
Midterm Exams
Below is a list of exam dates with the material covered for each.
Grading Scale
Let x represent your grade.
A: 90 ! x < 100
B: 80 ! x < 90
C: 70 ! x < 80
D: 60 ! x < 90
F: 0 ! x < 60
Exam 1: Wednesday, 2/23/15
§ Chapters 1-3, excluding Sections 3.3.3-3.7
Exam 2: Wednesday, 4/6/15
§ Chapter 4
Exam 3: (Take Home) Due Monday, 5/11/15
§ Chapters 7-8, excluding Sections 7.9 & 8.7
We will have 3 midterm exams: 2 in-class & 1 take home. You will have the
entire class period to complete an in-class exam. Calculator use may be allowed
by my discretion. The dates described are tentative to the pace of course and may
be subject to change. Please also note that you will need to bring a photo ID to
the first in-class exam.
I also reserve the right to change an in-class exam to being take home or add a
take home component for any given test. There will be no make-ups under any
circumstances. That said, I will drop the lowest midterm score from your final
grade.
Final
Date & Time: Wednesday, 5/20/15, 4:00 – 6:00 pm
Covers: Chapters 1-4 & 7-9, excluding Section 3.3.3-3.7, 7.9, & 8.7
The final exam will be cumulative. Calculator use may be allowed by my
discretion. Absolutely no make-ups or rescheduling of the final exam (especially
if you schedule a flight that conflicts with this date). You’ve been warned!
Projects
Throughout the course you will be assigned various projects of in depth problems
or material not covered in lecture. These may be individual or group based.
Some may entail short write-ups, while others may be more involved such as a
paper or presentation during office hours. Please see individual project
instructions for details.
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Course Help
Calculus is not any easy subject, which means the average student will need help
throughout the course. Below is a list of resources where students can go to get
course help.
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Office Hours
Forming a study (Do this within the first two weeks of the semester!)
Check out another Differential Equations textbook from the Library for
reference
Purchase the Student Solutions Manual for the textbook and/or
Schaum’s Outlines for Differential Equations
Khan Academy: www.khanacademy.org/math/differential-equations
Paul’s Notes: http://tutorial.math.lamar.edu/Classes/DE/DE.aspx
Students with Disabilities
Students with documented learning and/or physical disabilities may receive
reasonable classroom and/or testing accommodations. These arrangements need
to be made with me privately within the first two weeks of class or as soon as the
documentation is determined. You will have to show me an accommodation slip
that you received from DSPS. You can find DSPS in D-117 or contact them via
telephone at (510) 748-2328. Unfortunately, I will not be able to accommodate
last minute requests.
Religious Holidays
Reasonable accommodations will be made for you to observe religious holidays
when such observances require you to be absent from class activities. It is your
responsibility to inform me during the first two weeks of class, in writing, about
such holidays.
Academic Honesty
College of Alameda and myself take cheating very seriously. Please make sure
any work produced is your own. It is also your responsibility to ensure other
students are not copying from you (e.g., two identical tests will be considered
both students' responsibility). At the very least, anyone caught cheating will
receive a 0 on the assignment, and will NOT have any of their lowest scores
dropped from their final grade. Consequently, your grade will severely be
reduced. In some instances, a student may be asked to leave the course. You also
run the risk of being reported to the dean and facing disciplinary action from the
college. If you are having trouble in the course come talk to a counselor or me. I
am more than happy to help you with any concepts giving you trouble, or tips for
studying. Cheating is never worth risking your education!
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Tentative Calendar
Please note that any section listed with an (*) will be covered at home through a
reading assignment.
Week of Wednesday, 1/21/15
§ Syllabus & Introductions
§ Section 1.1: Background*
Week of Monday, 1/26/15
§ Section 1.2: Solutions and Initial Value Problems
§ Section 1.3: Direction Fields
§ Section 1.4: Euler’s Method
§ Section 2.1: Introduction Motion of a Falling Body*
Week of Monday, 2/2/15
§ Section 2.2: Separable Equations
§ Section 2.3: Linear Equations
§ Section 2.4: Exact Equations
Week of Monday, 2/9/15
§ Section 2.5: Special Integrating Factors
§ Section 2.6: Substitutions & Transformations
§ Section 3.1: Mathematical Modeling
§ Section 3.2: Compartmental Analysis
Week of Monday, 2/16/15
§ Monday: No Class – Washington’s Birthday
§ Wednesday: Section 3.2 (Continued)
§ Section 4.1: Introduction: The Mass-Spring System*
Week of Monday, 2/23/15
§ Monday: Section 4.2: Homogeneous Linear Equations: The General
Solution
§ Wednesday: Midterm 1
Monday, 3/2/15
§ Section 4.3: Auxiliary Equations with Complex Roots
§ Section 4.4: Nonhomogeneous Equations: The Method of
Undetermined Coefficients
§ Section 4.5: The Superposition Principle & Undetermined Coefficients
Monday, 3/9/15
§ Section 4.6: Variation of Parameters
§ Section 4.7: Variable Coefficient Equations
§ Section 4.8: Qualitative Considerations for Variable-Coefficient and
Nonlinear Equations
Week of Monday, 3/16/15
§ Section 4.9: A Closer Look at Free Mechanical Vibrations
§ Section 4.10: A Closer Look at Forced Mechanical Vibrations
§ Section 8.1: Power Series and Analytic Functions*
§ Section 8.2: Power Series & Analytic Functions*
Week of Monday, 3/23/15
§ Section 8.3: Power Series Solutions to Differential Equations
§ Section 8.5: Cauchy-Euler Equidimensional Equations
§ Section 8.6: Method of Frobenius
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Week of Monday, 3/30/15
§ No Class All Week – Spring Break
Week of Monday, 4/6/15
§ Monday: Section 8.8: Special Functions (Bessel’s Equation)
§ Wednesday: Midterm 2
§ Section 7.1: Introduction: A Mixing Problem*
Week of Monday, 4/13/15
§ Section 7.2: Definition of the Laplace Transform
§ Section 7.3: Properties of the Laplace Transform
§ Section 7.4: Inverse Laplace Transform
Week of Monday, 4/20/15
§ Section 7.5: Solving Initial Value Problems
§ Section 7.6: Transforms of Discontinuous and Periodic Functions
§ Section 9.2: Review 1: Linear Algebraic Equations*
§ Section 9.3: Review 2: Matrices and Vectors*
Week of Monday, 4/27/15
§ Section 7.7: Convolution
§ Section 7.8: Impulses and the Dirac Delta Function
§ Section 9.1: Introduction to Systems of Linear Differential Equations
Week of Monday, 5/4/15
§ Section 9.4: Linear Systems in Normal Form
§ Section 9.5: Homogeneous Linear Systems with Constant Coefficients
§ Section 9.6: Complex Eigenvalues
Week of Monday, 5/11/15
§ Section 9.7: Nonhomogeneous Linear Systems
§ Section 9.8: The Matrix Exponential Function
Week of Monday, 5/18/15
§ Monday: No Class
§ Wednesday: Final Exam (4:00 – 6:00 pm)
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