SPRING 2009
Instructor:
Dr. S. Ugwuoke
Phone: 546-8543
Office: Graphics Bldg. Room 4
E-mail: sugwuoke@shawu.edu
Office Hours: 11 AM - 1:00 PM, Monday to Friday, or by Appointment
Required Text
Edwards, C. Henry and Penny, David E.
Differential Equations and Boundary Value Problems; 3rd Edition; Copyright ©
2004 by Pearson Education, Inc.
Supplementary Texts
:Technology Used in This Course
1. Blackboard and Quizlab (websites)
2. Maple V Release 5 (Computer Lab)
3. Graphing Calculator (TI-92); 4. PcCalculator
Program Mission
The mission of the Mathematics Program is to prepare students with the knowledge, skills, and competencies, for employment in fields of work requiring quantitative and problem solving skills, and also to pursue graduate studies in Pure and Applied Mathematics. The mission is also to produce graduates who are equipped with analytical and critical thinking skills to enable them to formulate problems, solve them, and interpret their solutions, and communicate the solution.
Program Goals
The primary goals of the Mathematics unit for this period are as follows:
Program Learning Outcomes (PLO)
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| 1.1 Students will be able to draw the graphs of various functions, find their derivatives, integrals, identify some properties of functions, find the maximum and minimum values of functions using algebraic and calculus techniques. They will also be able to use numerical techniques to find definite integrals of functions and apply all these techniques to solve application problems. |
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| 1.2 Students will be able to represent a given data in diagrams, find various measures of central tendencies, dispersions, correlation between variables, and other statistical parameters. They will also be able to find probabilities of certain simple and compound events using various techniques of probability using probability distributions. Students will also be able to apply these techniques to solve application problems. |
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| 1.3 Students will be able to solve systems of linear equations, find the matrices representing linear transformations, do matrix computations. They will also be able to solve ordinary differential equations both algebraically and numerically. Students will be able to apply these techniques to solve application problems. |
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| 2.1 Students will be able to understand the various techniques of proving theorems and will be able to state and prove theorems using definitions and properties. |
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| 2.2 Students will be able to use differentiation and integration techniques to solve application problems in optimizing techniques for functions in Business, Economics, Sociology etc, also find areas and volumes of planes and solids using definite integrals, and multiple integrals. |
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| 2.3 Students will be able to use Eigen values, Eigen vectors in solving and predicting long range effects in other areas of study. Students will be able to solve application problems using these techniques; |
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| 3.1 Periodic meetings of all major students with all the math faculty will be arranged to give an opportunity to students and faculty to communicate |
| and exchange ideas to provide the students with what their academic needs are and make their learning experience more enjoyable. |
| 3.2 The advisors will meet with their advisees at least two times a semester to make sure the students are taking the right courses and are in the |
| right track for timely graduation. Also they will address any academic needs the advisees have and make them more comfortable to stay and |
| complete the major program. |
| 3.3 Inform the students of opportunities on Summer Internships and summer Institutes and encourage them to get these experiences and also take |
| them to undergraduate conferences where they can meet other undergraduate math students and exchange ideas and learn about graduate school |
| and research opportunities. Make arrangements for organizing tutoring sessions for students who need help in their class work. |
General
Description
This is a study of ordinary differential
equations of different orders and degrees, their solutions, meanings and methods
of solutions, trajectories, numerical techniques, Laplace transforms, power
series methods, systems of equations, and applications to other disciplines and
real-life problems.
Student Learning Outcomes
(SLO)| At the completion of this course students should be able to do the following: | PLO-s linked to the SLO-s | Assessment tools for Student learning Outcomes |
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· demonstrate an understanding of basic concepts of ODE such as |
1.3 | Quizzes,Assignments, Tests, & Projects |
| order and degree of ODE, linear DE., notations; | ||
| · distinguish between a solution, a particular solution, and | ||
| a general solution of ODE; | ||
| · classify first order differential equations; | ||
| · solve separable first-order differential equations, including | ||
| initial value problems; | ||
| · solve homogeneous first-order DE; · solve exact first-order DE; | ||
| · find solution of DE by using integrating factors; | ||
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· solve linear first-order DE; · apply first-order DE to real-life situations; |
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| · identify and solve linear differential equations; | ||
| · solve second-order linear homogeneous equations with constant | ||
| coefficients by using characteristic equations; | ||
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· solve nth order linear homogeneous equations with constant |
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| coefficients by using characteristic equations; | ||
| · solve ODE using the method of undetermined coefficients; | ||
| · solve initial-value problems; | ||
| · apply second-order linear DE with constant coefficients; | ||
| · solve problems involving Laplace transforms; | ||
| · apply Laplace transforms and its inverse in solving DE; | ||
| · solve systems of linear differential equations; | ||
| · apply systems of linear DE to real-life situations; | ||
| · apply numerical methods, e.g. Euler method, in solving DE. | ||
| · demonstrate an understanding of power series solution of D.E. | ||
| · demonstrate an understanding of analytical, qualitative, and | ||
| numerical techniques of making predictions. |
Topic Outline
Chapter:
2.Mathematical Models and Numerical Methods
2.1 Population Models2.2 Equilibrium Solutions and Stability
2.3 Acceleration-Velocity Models
2.4 Numerical Approximation: Euler's Method2.5 A closer look at the Euler's Methods
2.6 The Runge Kutta Method
3.Lnear Equations of Higher Order
3.1 Introduction: Second-Order Equations3.2 General Solutions of Linear Equations
3.3 Homogeneous Equations with Constant Coefficients
3.4 Mechanical Vibrations
3.5 Non-homogeneous Equations and Undetermined Coefficients3.7 Electrical Circuits
4.Introduction to Systems of Differential Equations;
4.1 First-Order Systems and Applications4.2 The Method of Elimination
4.3 Numerical Methods for Systems
5.Linear Systems of Differential Equations
5.1 Matrices and Linear Systems
5.2 The Eigenvalue Method for Homogeneous Systems
5.3 Second–Order Systems and Mechanical Applications
5.4 Multiple Eigenvalue Solutions
5.5 Matrix Exponentials and Linear Systems
5.6 Non-homogeneous Linear Systems
6.Nonlinear Systems and Phenomena
6.1 Stability and the Phase Plane
6.2 Linear and Almost Linear Systems
6.3 Ecological Models: Predators and Competitors
6.4 Nonlinear Mechanical Systems
6.5 Chaos in Dynamical Systems
7.Laplace Transform Methods
7.1 Laplace Transforms and Inverse Transforms
7.2 Transformation of Initial-Value Problems
7.3 Translation and Partial Fractions
7.4 Derivatives, Integrals, and Products of Transforms
7.5 Periodic and Piecewise Continuos Input Functions
7.6 Impulses and Delta Functions
8.Power Series Methods
8.1 Power Series Methods
8.2 Series Solutions Near Original Points
8.3 Regular Singular Points
8.4 Methods of Frobenius: The Exceptional Cases
8.5 Bessel’s Equation
8.6 Applications of Bessel Functions
9. Fourier Series Methods
9.1 Periodic Functions and Trigonometric Series
9.2 General Fourier Series and Convergence
9.3 Fourier Sine and Cosine Series
9.4 Applications of Fourier Series
9.5 Heat Conduction and Separation of Variables
9.6 Vibrating Strings and One-Dimensional Wave Equation
9.7 Steady-State Temperature and Laplace’s Equation
10.Eigenvalues and Boundary-Value Problems
10.1 Sturm-Liouville Problems and Eigenfunction Expansions10.2 Applications of Eigenfunction Series
10.3 Steady Oeriodic Solutions and Natural Frequences
10.4 Cylindrical Coordinate Problems
Alignment With Standards
NCDPI 3.1, 3.4, 3.5
Grading System
Student Classroom Decorum Expectations
To enhance the learning atmosphere of the classroom, students are expected to dress and behave in a fashion conducive to learning in the classroom. More specifically, students will refrain from disruptive classroom behavior (i. e., talking to classmates, disrespectful responses to teacher instructions; swearing; wearing clothes that impede academic learning such as but not limited to, wearing body-revealing clothing and excessively baggy pants; hats/caps, and/or headdress). Students will turn off telephones prior to entering the classroom.
Students who exhibit the behaviors described above, or similar behaviors will be immediately dismissed from class at the third documented offense. The student will be readmitted to class only following a decision by the department chair. The student may appeal the decision of the department chair to the Dean of the College offering the course, and, subsequently, to the Office of the Vice President for Academic Affairs, and then to the President of Shaw University. The decision of the President will be final.
Failure to follow the procedures herein outlined will result in termination of the appeal, and revert to the decision of the department chair.
Each behavior construed by the teacher/professor as noncontributive to learning will be recorded, properly documented, and appropriately reported to the student and to the chair of the academic department offering the course. The report will be in written form with a copy provided to both the student and the department chair. The faculty member should retain a copy for his/her own records.
Additional student behavior codes may be found in Student Affairs.