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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