Math 4            Differential Equations (2 credits), Linear Algebra  (1 credit), and

                        Probability and Statistics  (2 credits)

Pre-Requisites:  Multivariable Calculus, Introduction to Linear Algebra (or Math 1, 2 and 3)

                         

Course Description:

This course begins with applications involving the solution of second order linear differential equations with constant coefficients. Such equations are then viewed in their relationship to selected partial differential equations with both initial and boundary values conditions. Second order equations are extended to systems of linear differential equations, which are also seen in their relation to systems of difference equations. These are solved with a formal linear algebra approach using eigenvalues and eigenvectors and then extended into nonlinear systems.  Vector spaces, spanning sets, and, linear transformations with corresponding null spaces are studied. The last portion of the class is spent on probability and statistics with an emphasis on exploratory data analysis and applications involving selected distributions and reliability analysis. The class meets once week in the computer laboratory. The primary software used is Mathematica, and SPSS and Excel are used for the statistics portion. It is a required course for all mathematics majors and minors and all engineering majors.

Course Objectives:  Students Will

·        Model situations that lead to second order linear differential equations with constant coefficients, then solve and interpret these equations numerically and analytically

·        Assimilate topics in multivariable calculus, differential equations, and Fourier series by exploring the solution of selected partial differential equations

·        Build on experience using linear algebra to solve systems of difference equations to introduce modeling and solution of systems of linear differential equations

·        Use slope fields and solution curves in the phase plane to predict the behavior of solutions near equilibrium points and to relate the stability of the equilibrium points to the eigenvalues and eigenvectors of the coefficient or Jacobian matrix

·        Understand the concept of a vector space and its spanning sets

·        Understand mappings and linear transformations and the relationship between the eigensystem and the kernel and image

·        Develop an understanding of the basic principles of probability and statistics

·        Experience the role of probability and statistics in a wide variety of applications

Grading Policies:

Exams are given every two to three weeks. Labs are done every week.

Relative weights:            Exams (65%), Homework (10%), Projects/Labs (25%)

                                    Projects are computer intensive and students are encouraged to work in pairs.

Some projects, including most parts of the Lake Pollution Project, results will be presented orally as well as in written form.

Required Texts:  Two Sets of Examples

Example 1:            Differential Equations and Linear Algebra  by Goode, © 2000 Prentice Hall

Modern Engineering and Statistics by Lapin , © 1997 Prentice Hall

Example 2:            Differential Equations and Linear Algebra  by Edwards & Penney, © 2001 Prentice Hall

Probability and Statistics by Hastings , © 1997, Addison-Wesley-Longman

Course Outline

Meeting 5 Days Per Week for 14 Weeks

Topics

Chapters in Texts

Goode

Edwards

Penney

Second Order Linear Differential Equations with Constant Coefficients (2 weeks)

    
 

Homogeneous – Modeling and Solving

Ch 2

Ch 8
 

Nonhomogeneous – Undetermined Coefficients

Ch 2

Ch 8
 

Bungee Jumping Project

BJ ILAP

BJ ILAP

Boundary Value Problems and a Partial Differential Equation        (1 week)

    
 

Heat Equation – Modeled and Numerically Solved, Analytical  Solution with Eigenfunctions, Eigenvalues, and Fourier Series

Supplement

Supplement

Systems of Difference and Differential Equations           (3 weeks)

    
 

Motivation for Eigenvalues and Eigenvectors through Lake Pollution ILAP Parts I & 2

LP ILAP

LP ILAP
 

Modeling and Solving Linear Systems of Homogeneous Differential Equations

Ch 8

Ch 7
 

Analysis of Nature of Solutions and Stability of Equilibrium Points Using Phase Plane Slope Fields and Eigenvalues and Eigenvectors

Ch 8

Ch 7
 

Modeling, Linearizing, and Numerically Solving Systems of Nonlinear Differential Equations

Ch 8

Ch 9

Vector Spaces and Transformations  (2 weeks)

    
 

Vector Spaces, Spanning Sets, Linear Inependence

Ch 5

Ch 4
 

Linear Transformations, Mappings, and Kernels

Ch 6

Supplement
 

Relationship Between Null Space and Image Space and Eigensystem

Ch 6

Supplement
 

Leslie Matrices with Harvesting and Eigensystem Analysis

Worksheet

Worksheet
       

Topics

Lapin

Hastings

Statistics (1.5 week)

    
 

Overview of Sampling and Design of an Experiment

Ch 1

Ch 7
 

Exploratory Data Analysis and Measures of Location & Dispersion

Ch 2

Ch 7
 

Statistical Process Control and Sampling Distributions

Ch 3

Ch 6
 

Least Squares Linear Curve Fitting and Regression

Ch 4

Ch 10
 

Nonlinear Curve Fitting

Ch 5

Worksheet

Probability (3 weeks)

    
 

General Overview, Tree Diagrams, and Reliability

Ch 6

Chs 1, 4
 

Markov Models as a Linear Transformation

LP ILAP - 4

LP ILAP - 4
 

Decision Analysis

Worksheet

Worksheet
 

Probability Density Functions and Cumulative Distribution Functions and Parameters

Ch 7

Ch 1
 

Selected Discrete and Continuous Distributions

Chs 7, 8

Chs 2, 3

Inferential Statistics (1.5 weeks)

    
 

Fitting Data Patterns with Distributions

LP ILAP - 5

LP ILAP - 5
 

Confidence Intervals and Introduction to  Hypothesis Testing

LP ILAP - 6

LP ILAP - 6
 

Simulation Project (for example, a baseball game)

Worksheet

Ch 5