Math 3740                                                                            Spring 2010

Prerequisites by topics:

1. Calculus I and II - two-semester sequence in differential and integral calculus
2. Multivariate Calculus - one-semester course in multivariable calculus
3. Elements of Matrix Algebra from a multivariable calculus course

Academic Integrity: Students are responsible for making themselve aware of and understanding the policies and procedures in the Undergraduate (pp. 274-276) [Graduate (pp.25-27)] Catalog that pertain to Academic Honesty. These policies include cheating, fabrication, falsification and forgery, multiple submission,
plagiarism, complicity and computer misuse. If there is reason to believe you have been involved in academic dishonesty, you will be referred to the Office of Student Conduct. You will be given the opportunity to review the charge(s). If you believe you are not responsible, you will have the opportunity for a hearing. You should consult with me if you are uncertain about an issue of academic honesty prior to the submission of an assignment
or test.

Objectives:

1. Understand differential equations as an important tool for modeling of physical and engineering processes.
2. Understand symbolic and numerical methods for finding solutions of differential equations and for analysis of their behavior.
3. Understand structure of solutions of linear systems of differential equations and related concepts of linear algebra (linear algebraic systems of equations, eigenvalues and eigenvectors)
4. Understand Laplace transform methods for finding solutions of linear differential equations
5. Understand concepts of stability of equilibrium solutions of nonlinear systems and methods of linearization of nonlinear systems near equilibrium
6. Understand the possibilities of modern computer algebra systems (Maple) in analysis of differential equations and their solutions, numerical methods and visualization for solutions.
7. Improve problem-solving skills.

Topics:

1. The first-order differential equations, uniqueness and existence of solutions, symbolic methods of integration
2. Applications of differential equations for modeling of population dynamics and motion
3. Numerical methods for first-order differential equations: Euler, improved Euler and Runge-Kutta methods
4. Matrices, linear system s of algebraic equations and their solutions, vector spaces, bases and linear independence
5. Linear differential equations of higher order, general solutions, superposition principle
6. Homogeneous linear equations with constant coefficients, their solutions
7. Nonhomogeneous linear equations - methods of undetermined coefficients and variation of parameters
8. Applications of linear differential equations for modeling of mechanical vibrations and electric circuits
9. Linear first-order systems of differential equations, existence and uniqueness of solutions
10. The eigenvalue method for linear systems with constant coefficients
11. Matrix exponentials and linear systems, fundamental matrix solutions
12. Nonhomogeneous linear systems
13. Numerical methods for first-order nonlinear systems of differential equations
14. Equilibrium solutions of nonlinear systems. Stability and asymptotic stability of equilibrium solutions. Phase plane
15. Linearization of nonlinear systems near equilibrium. Mechanical applications
16. Laplace transform method for solving linear differential equations
17. Power series solution techniques on example of Bessel's equation

These topics correspond to Chapters 1-5 and selected sections of Chapters 6-7 and 9-10 of the text. Students are responsible for all material in the text and all material presented in class. This includes any material not in the text and all material in the text that was not presented in class.

Calculator and Maple:

A graphing calculator is required for this class. A TI-89 or TI-92 PLUS is recommended. We will also systematically use computer algebra system Maple in this course for visualization, demonstration and analysis of behavior of solutions of differential equations. Students learn applications of Maple tools for visualization of slope fields, for finding explicit solutions of some classes of differential equations, for using procedures for numerical solution of differential equations. Computer projects can include such topics as visual study of stability of equilibrium solutions of population dynamics equations, development of numerical procedure based on Runge-Kutta method for solution of some ballistics equations, study of resonance frequencies for some mechanical systems, phase portraits of two-dimensional nonlinear systems. Click here for some sample maple worksheets.

Performance Objectives:

1. Students will be able to understand a concept of solution of differential equations, an importance of the fact of its existence and uniqueness in modeling of physical and dynamical processes.
2. Students will be able to find explicit solutions for some classes of first-order differential equations using methods of separation of variables, change of variables, integrating factor.
3. Students will demonstrate ability to use differential equations in modeling of some dynamical processes: population dynamics, motion under resistance forces, and oscillations in mechanical and electrical systems.
4. Students will be required to use a variety of numerical methods for solving differential equations, to know their relative advantages and disadvantages, to provide error analysis of numerical procedures.
5. Students will be required to know main concepts of matrix and linear algebra, to be able to use Gauss-Jordan elimination method for solving linear systems of algebraic equations.
6. Students will be able to explain a structure of general solutions of linear differential equation of higher order, to find general solution of linear equation with constant coefficients, to find solution of nonhomogeneous linear equation.
7. Students will be able to explain a general structure of first-order linear system of differential equations, to use the eigenvalue method to find general solution of linear system with constant coefficients, to use numerical methods for solving first-order nonlinear systems.
8. Students will be required to know matrix solutions of linear systems, including matrix exponentials, and to use them for finding solutions of nonhomogeneous linear systems.
9. Student will be able to understand concepts of phase plane for nonlinear system, equilibrium solutions and their stability, to use linearization method for stability analysis of equilibrium solutions.
10. Student will be able to understand a concept of Laplace transform, to use it for solving linear equations and linear systems
11. Student will be able to use a modern computer algebra system such as Maple for analysis of behavior of solutions of differential equations

Homework:

Homework assignments and a tentative schedule are give in the table blow. Although none of the homework will be collected, you are responsible for all of the problems. If you have any questions about problems, please ask them in class or in office hours.

Quizzes:

A total of 6 quizzes will be given. They will cover all of the material before the day of the quiz. No make up quiz will be given. However, I will drop your worst quiz.

Final:

The final exam will be 5:00--7:00pm on Monday, April 26.

Evaluation:

The final is 30%, computer projects is 10% and the quizzes count for 60%. This gives a total of 100%.
Grading scale is approximately as follows:
                                    A (86-100%) BA(79-85.99%) B (71-78.99%) CB(63-70.99%)
                                    C(55-62.99%) DC(50-54.99%) D(43-49.99%) E(0-42.99%) 

Tentative schedule and homework assignments: