MTH 132 (sec 104) Syllabus Fall 2004
IST 231 (sec 201) Syllabus Spring 2009
CRN 2874
Prerequisites: Completion of IST230 with a grade of C or higher
Learning Objectives: To understand the theory and practice of solving differential equations.
Meeting time : Lectures T R 12:30 -1:45 pm Rm 119 Morrow Library
Lab R 4 - 4:50 pm Rm 119 Morrow Library
Instructor : Dr. Alan Horwitz Office : Room 737 Smith Hall
Phone : (304)696-3046 Email : horwitz@marshall.edu
Text : Differential Equations and Linear Algebra, 2nd edition, Farrow, Hall, McDill, West Prentice Hall
Software : STELLA, Matlab, Mathematica
Grading : attendance 6% ( 33 points )
homework 18% (100 points)
lab assignments 18% (100 points)
2 major exams 36% (200 points)
( if there’s a 3rd exam, I’ll use the highest two grades )
final( comprehensive ) exam 22% (122 points)
555 points total
Final exam date : Tuesday May 5, 2009 from 12:45-2:45 pm
General Policies :
Attendance at lecture and labs is required and you must bring your text and graphing calculator . Lab sessions may occasionally be used for lectures if we need the extra time to cover material. You are responsible for reading the text, working the exercises, coming to office hours for help when you’re stuck, and being aware of the dates for the major exams as they are announced. The TI-83 will be occasionally used in classroom demonstrations, although it is not useful for calculus operations. You may use any brand you wish (although I may not be knowledgeable on its operation).
Major exams are scheduled for approximately the 6th and 11th weeks, with the exact dates announced at least one week in advance. Exams will cover material from the lectures and assigned ( both graded and uncollected ) homework exercises. You will be expected to show all of your work on the questions and grading will follow a standard scale( A 90-100, B 80-89, C 70-79, etc.) or may be curved slightly.
Makeup exams will be given only if you have an acceptable written excuse with evidence and/or you have obtained my prior permission. I don’t like to give makeup exams, so don’t make a habit of requesting them. Makeups can possibly be more difficult than the original exam and must be taken within one calendar week of the original exam date, i.e. if the exam was on Thursday, then you must take the makeup before the next Thursday). You can’t make up a makeup exam: if you miss your appointment for a makeup exam, then you’ll get a score of 0 on the exam. If you anticipate being absent from an exam due to a prior commitment, let me know in advance so we can schedule a makeup. If you cannot take an exam due to sudden circumstances, you must call my office and leave a message or e-mail me on or before the day of the exam!
Your homework grade will be based on your highest 10 homework scores. Your lab grade will be based on all of your lab scores. The homework and lab assignments must be turned in on time and should reflect your own work and thinking , not that of your classmates. If there are n homework or lab assignments which appear to be identical ( where n represents a positive integer), then I will grade with one score, which will be divided by n to give as the score for each assignment. For example, if three students submit identical assignments and the work gets a score of 9,
then each assignment will get a score of 3.
In borderline cases, your final grade can be influenced by factors such as your record of attendance, whether or not your exam scores have been improving during the semester, and your class participation.
Attendance Policy : This is NOT a distance learning course !
Attendance is 6% of your grade( 33 points total). If your grade is borderline, these points can be important
in determining the final result. Everyone starts out with 33 points, then loses 2 points for each lecture or lab period missed.
Your attendance score will be graded on a stricter curve than your exam scores. Attendance will be checked daily
with a sign-in sheet. Signing for someone other than yourself will result in severe penalties!!
Having more than 3 weeks worth of unexcused absences (i.e., 9 of 43 lecture and lab periods combined ) will automatically result in a course grade of F! Absences which can be excused include illness, emergencies, or official participation in another university activity. Documentation from an outside source ( eg. coach, doctor, court clerk…) must be provided. If you lack documentation, then I can choose whether or not to excuse your absence. Being habitually late to class will count as an unexcused absence for each occurrence. Carrying on conversations with your neighbor or sleeping through the lecture on a regular basis could be counted as an unexcused absence. Leaving early without permission will be regarded as an unexcused absence. If you must leave class early for doctor’s appointment , etc., let me know at the beginning and I’ll usually be happy to give permission.
IST 231 (sec 201) Syllabus Spring 2009
CRN 2874
( continued )
HEED THIS WARNING:
Previously excused absences without documentation can, later, instantly change into the unexcused type if you accumulate an excessive number ( eg. more than 2 weeks worth ) of absences of any kind, both documented and undocumented :
You are responsible for keeping track of the number of times you’ve been absent. I won’t tell you when you’ve reached the threshold..
Policy on Cap Visors :
During quizzes and exams, all cap visors will be worn backward so that I can verify that your eyes aren’t roaming to your neighbor’s paper.
Cell Phone and Pager Policy : Shut them off !
Unless you are a secret service agent, fireman, or paramedic on call, all electronic communication devices such as pagers and cell phones should be shut off during class.
The following brisk schedule optimistically assumes we will cover a multitude of topics at a rapid pace:
approximately two sections per week! Realistically speaking, we may surge ahead or fall somewhat behind,
but we can’t afford to fall too far off the pace. There may be variations from the order listed below in which material is covered. The two major exams will be approximately during the 6thand 11th weeks, plus or minus one week. Their precise dates will be announced at least one week in advance and the topics will be specified ( and may possibly differ from what is indicated below).
|Week |Dates |Approximate schedule: |
| |Spring 2009 |Sections Covered |
|1 |1/12 - 1/16 | 1.1, 1.2, 1.3 |
|2 |1/20 - 1/23 | |
| |( MLK day on 1/19 ) |1.4 |
|3 |1/26 - 1/30 | 1.5, 2.1 |
|4 |2/2 - 2/6 | 2.2, 2.3 |
|5 |2/9 - 2/13 | 2.4, 2.5 |
|6 |2/16 - 2/20 | |
| | |Exam 1, 3.1, 3.2 |
|7 |2/23 - 2/27 |3.3, 3.4 |
|8 |3/2 - 3/6 |3.5, 3.6, 4.1 |
|9 |3/9 - 3/13 |4.1, 4.2, 4.3 |
|10 |3/16 - 3/20 |4.3, 4.4 |
| |(3/20 is | |
| |last day to drop, | |
| |Spring Vacation | |
| |next week ) | |
|Week |Dates |Approximate schedule: |
| |Spring 2009 |Sections Covered |
|11 |3/30 - 4/3 |Exam 2, 5.1, 5.2 |
|12 |4/6 - 4/10 |5.2, 5.3 |
|13 |4/13 - 4/17 |6.1, 6.2 |
| | | |
| | | |
|14 |4/20 - 4/24 |6.2, 6.3 |
| | | |
| | | |
| | | |
|15 |4/27 - 5/1 |6.3, 6.5 |
| |(WEEK OF THE DEAD ) | |
| | | |
| |(Final Exam on | |
| |Saturday May 2) | |
Topics
* sections aren't in the formal syllabus, but we may need to use some material from them
1.1 Ordinary vs. partial differential equations
Order of a differential equation, dependent vs. independent variable
Examples of 1st order differential equation models:exponential growth/decay,
logistic growth, Newton’s Law of Cooling/Heating
Examples of 2nd order differential equation models:
Newton’s 1st Law of Motion combined with Hooke’s Law
Writing a 2nd order differential equation as a system of two 1st order equations
1.2 Verifying a function is a solution to a differential equation(DE)
Finding families of solutions for [pic] and solving initial value problems(IVP)
Sketching direction fields for a 1st order DE [pic], using them
to sketch solution curves to an IVP
Using isoclines to help hand-sketch direction fields
Analyzing concavity of solution curves to a 1st order DE
Identifying equilibrium solutions, deciding if they are stable, semistable or unstable
Identifying isocline curves
1.3 Solving separable DEs and writing the solutions in implicit or explicit form
Examples of solving separable IVPs
1.4 Using Euler’s Method to construct the Euler-approximate solution to a 1st order IVPs
Comparing the error between the Euler-approximate and true solution function
Roundoff errors
Calculating the local and global discretization errors
Advantages of second order and fourth order Runge-Kutta methods
Variable step versions of Euler’s Method
1.5 Applying Picard’s Existence and Uniqueness Theorem to IVPs
IST 231 (sec 201) Syllabus Spring 2009
CRN 2874
( continued ) Topics
2.1 Linear algebraic equations, non-homogeneous and homogeneous types
Recognizing linear vs. non-linear differential equations
Properties of linear operators
Being able to write linear algebraic equations using linear operators
Being able to write linear differential equations using linear operators
Superposition Principle for linear homogeneous equations, algebraic and differential types
The Non-Homogeneous Principle for linear non-homogeneous equations
Method of finding all solutions to a non-homogeneous linear DE
2.2 Using integrating factors to solve 1st order non-homogeneous linear DEs
Viewing the general solution as a sum of a steady state solution and transient part
2.3 Using separation of variables to solve an exponential growth/decay DE
Half life and doubling time in decay/growth models
Continuous compounding of interest
2.4 Solving DEs for mixing models
Newton’s Law of Cooling problems
2.5 Using direction fields to sketch solution curves of non-linear 1st order DEs,
identifying isoclines, where slopes are positive/negative, and equilibrium solutions
Using phase line analysis to sketch behavior of solutions to an autonomous 1st order DE
Growth models with a variable growth rate
Using separation of variables and partial fractions to solve logistic equations
Intrinsic growth rate and carrying capacity
Solutions to the threshold equation and the importance of threshold level
2.6* Verifying a function is a solution to a system of differential equations
Recognizing systems of coupled/decoupled DEs
Structure of an autonomous system of two 1st order DEs with an initial condition
Identifying vector fields and trajectories(solution curves) in the phase plane for
an autonomous system of two 1st order DEs
Stable vs. unstable equilibrium points of a system
Vertical nullclines vs. horizontal nullclines of a system
Hand sketching trajectories by determining directions of vector fields
in regions between vertical and horizontal nullclines
Lotka-Volterra system for predator-prey model
3.1 Identifying types of matrices, including zero matrices, diagonal matrices and identity matrices
Algebraic properties of addition and scalar multiplication of matrices
Viewing row and column vectors as matrices
Computing the scalar product of a row vector by a column vector
Computing the length of a vector
Compatibility condition for multiplying two matrices together
Computing the product of two compatible matrices
Matrix multiplication is not commutative
Algebraic properties of the transpose of a matrix
Rules for differentiating a matrix whose entries are functions
IST 231 (sec 201) Syllabus Spring 2009
CRN 2874
( continued ) Topics
3.2 Geometric interpretations of systems of equations
Rewriting a system of linear equations as a matrix equation, identifying the coefficient matrix
Writing the augmented matrix of a system of linear equations
Performing elementary row operations on an augmented matrix
Using pivot columns and Gauss Jordan Reduction to put an
augmented matrix into Row Reduced Echelon Form
Reading the solutions to a system from row reduced echelon form:
recognizing when the system is inconsistent, when parameters are needed, when
the solution is unique
Applying the Superposition and Non-Homogeneous Principles to systems of linear equations
Determining rank of a row reduced echelon form matrix
3.3 Checking if one matrix is the multiplicative inverse of another matrix
Transpose of an inverse matrix is the inverse of the transpose
Gauss Jordan Reduction to solve more than one system with the same coefficient matrix,
but different right hand sides
Gauss Jordan Reduction method of finding the inverse matrix, if one exists
Using an inverse matrix to solve a system with an invertible coefficient matrix
3.4 Calculating the determinant of an [pic]matrix by hand using minors and cofactors
Properties of determinants related to products, transposes, and row operations on matrices
A square matrix is invertible if and only if its determinant is nonzero
Using Cramer’s Rule to solve systems with an invertible coefficient matrix
Method of Least Squares explained as solving a system of linear equations
3.5 Properties of vector spaces
Examples of vector spaces and vector subspaces, including solution spaces to homogeneous DEs
and spaces of polynomial functions
3.6 Linear combinations of vectors
Span of a set of vectors and the Span Theorem
Standard basis vectors in [pic]
Using a matrix equation to determine if a vector in [pic] lies in the span of other vectors
Determining whether a set of vectors is linearly independent or linearly dependent
Checking linear independence of functions
Using the Wronskian Test for linear independence of differentiable functions
Definition of basis and dimension of a vector space
Checking whether a collection of vectors is a basis for a vector space
Finding the dimension of the column space of a matrix
4.1 Hooke’s Law and damping force
2nd order DE for a damped harmonic oscillator, both unforced and forced
General solutions to undamped, unforced harmonic oscillator DE
Converting a solution to Alternate Form and identifying amplitude,
phase angle and phase shift, circular and natural frequencies, period
Converting a 2nd order DE into a system of two 1st order DEs
Graphing in the phase plane the phase portrait of solutions to autonomous 2nd order DEs
Modeling LRC circuits with 2nd order DEs
IST 231 (sec 201) Syllabus Spring 2009
CRN 2874
( continued ) Topics
4.2 Characteristic equation and characteristic roots of a 2nd order constant coefficient linear DE
Solutions when the characteristic equation has distinct real roots or repeated real roots
Overdamped and critically damped spring-mass systems
Existence and uniqueness of solutions to an IVP for a 2nd and nth order homogeneous linear DE
Basis for solution space of 2nd and nth order homogeneous DE has
2 linearly independent solutions
Wronskian Test for independence and dependence on solutions
to nth order homogeneous linear DE
4.3 Basis for solution space for a 2nd order homogeneous linear DE when the
characteristic roots are complex conjugates
Solutions in regular and alternate form for an underdamped spring-mass systems
Using characteristic roots to find general solutions of nth order homogeneous linear DEs
4.4* Using operator notation to write a linear differential equation
Superposition Principle for Non-homogeneous linear DEs
Non-homogeneous Principle for linear DEs
Solving non-homogeneous DEs by the Method of Undetermined Coefficients when the
forcing function is a polynomial, exponential function, cosine or sine function or product of these
4.5* Solving 2nd order linear differential equations by Variation of Parameters
4.6* Solutions to forced spring-mass systems with no damping
Resonance occurs when forcing frequency matches natural frequency
4.7* Converting 2nd and higher order linear DEs into systems of 1st order DEs, and writing the
system as a matrix equation
5.1 Definition of a linear transformation and examples, including rotations and projections
Image of a linear transformation
Finding standard matrix of size [pic]associated with a linear transformation from [pic]
5.2 One to one(injective) and onto(surjective) functions
Image Theorem
Rank of a linear transformation is the dimension of the image space
Using row reduced echelon form to find rank of a matrix multiplication operator
and basis of image
Kernel of a linear transformation with examples
Using row reduced echelon form to find basis for kernel of a matrix multiplication operator
Kernel Theorem , Dimension Theorem
Nonhomogeneous Principle for linear transformation
5.3 Eigenvectors and eigenvalues of a linear transformation
Characteristic equation of a matrix, the eigenvalues are roots of the characteristic polynomial
Finding eigenvalues of triangular matrices
Eigenspace Theorem, Distinct Eigenvalue Theorem
Finding the eigenspace of a repeated eigenvalue
Example of non-real eigenvalues: rotation transformation in the plane
Properties relating eigenvalues to their matrices
Finding eigenvectors and eigenvalues of a linear differential operator
5.4* Using column vector notation to represent a vector with respect to a given basis
Finding a change of basis matrix from a given basis to a standard basis and vice versa
Diagonalization Theorem
Using a basis of eigenvectors to diagonalize a matrix
When a matrix cannot be diagonalized
IST 231 (sec 201) Syllabus Spring 2009
CRN 2874
( continued ) Topics
6.1 Writing a linear 1st order DE system as a matrix-vector equation
Homogeneous linear 1st order DE systems
Existence and uniqueness of solutions to an IVP for a linear 1st order DE system
Superposition Principle for solutions to homogeneous linear DE systems
Solution Space Theorem: Solutions to homogeneous linear DE systems form a vector space
Solution Theorem on writing the general solution to a homogeneous linear DE system
in terms of linearly independent solutions
Writing the Fundamental Matrix of linear DE system
6.2 Converting a 2nd order linear DE into a system of two 1st order DEs and writing
as a matrix equation
Using eigenvectors to help solve a [pic] linear DE system with constant coefficients
when the coefficient matrix has two distinct real eigenvalues
Phase portraits of 2 dimensional linear DE systems:
finding a trajectory through a specified point by using initial conditions in
the general solution vector, sketching eigenvectors and the separatrix,
identifying equilibrium points ( stable,unstable or saddle types)
How sign of eigenvalues affects whether a trajectory moves
toward or away from an equilibrium point
How “speed” of the separatrix trajectory depends on magnitudes of the associated eigenvector
and how it affects the shape of nearby trajectories
Solving [pic]linear DE systems whose coefficient matrices have repeated real eigenvalues
and 2 linearly independent eigenvectors
Using generalized eigenvectors to help solve a [pic]linear DE system when the coefficient matrix
has less than two independent eigenvectors
Solving [pic]linear DE systems
Applying linear DE systems to two tank mixing problems
6.3 For a [pic] linear DE system with a real coefficient matrix,
complex eigenvalues and their eigenvectors come in conjugate pairs
Solving a [pic] linear DE system with complex conjugate eigenvalues by
splitting the solution vector into real and imaginary parts which use
exponential and cosine/sine functions
Sketching phase portraits for a [pic] linear DE system, including the
vertical nullclines and horizontal nullclines
Representing a solution vector as a product of an exponential “expansion” function,
a rotation matrix, and a column vector whose components are real and imaginary
parts of the eigenvector ; predicting the phase portrait from this representation
Solving a 2nd order DE by converting to a [pic] linear DE system
6.5
IST 231 (sec 201) Syllabus Spring 2009
CRN 2874
( continued )
Keeping Records of Your Grades and Computing Your Score
|Homework # |1 |2 |3 |
|score | | | |
Exam Total = sum of the two highest exam scores(not including final)
|grade curve for |Exam 1 |Exam 2 |Exam 3 |average of range values |
| | | | |for all three exams |
| A | | | | |
| B | | | | |
| C | | | | |
| D | | | | |
Absence # |1 |2 |3 |4 |5 |6 |7 |8 |9 |10 |11 |12 |13 |14 |15 |16 |17 |18 | |Date absent | | | | | | | | | | | | | | | | | | | |Excused? Y or N? | | | | | | | | | | | | | | | | | | | |Attendance Score |31 |29 |27 |25 |23 |21 |19 |17 |15 |13 |11 |9 |7 |5 |3 |1 |0 |0 | |
Attendance Score = 33 – [pic](# of days you were absent or extremely late)
Total % of Points=( Attendance Score
+ Adjusted Homework Score
+ Adjusted Lab Score
+ Exam Total
+ Final Exam Score)/555
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