M345 - Section 1 Spring 2009
Ordinary Differential Equations

Instructor      Dr. Iuliana Oprea,  Weber 123
Office Hours:  Mo 12-1PM, Tu 3-4pm and by appt., Phone: 491-6751 Office
Email:              juliana"at"math.colostate.edu; www: http://www.math.colostate.edu/~juliana/M345.html
Class Time and Room:  MTWF 2:00-2:50PM in EE 203

The general course page for M340: M340, sections 1-6

 Syllabus Homework

Lab Section: This course is formally split in a lecture and a lab session, which in practice will not be separated.

Required Textbook:
J. Polking, A. Boggess, D. Arnold: Differential Equations (2nd edition), Prentice Hall 2006, 2001, ISBN 0-13-143738-0

Course Objectives: The construction of mathematical models to address real-world problems is one of the most important aspects of each of the branches of science. It is often the case that these mathematical models are formulated in terms of equations involving functions as well as their derivatives, called differential equations. When only one derivative is involved, they are called ordinary differential equations - ODEs. The course will demonstrate the usefulness of ODEs for modelling physical, biological and other phenomena. Complementary mathematical approaches for their solutions will be presented, including analytical methods, graphical analysis and numerical techniques.

Synopsys: First order equations, mathematical models, linear equations of second order, the Laplace Transform, linear systems of arbitrary order and matrices, nonlinear systems and phase plane analysis, numerical methods.

Homework: Homework is collected at the beginning of every Wednesday lecture and is returned the next lecture. Late homework is not accepted.

Examinations: There will be two  in-class exams, on March 3 and April 14     and a Final Exam

Grading: Graded Homework, quizzes: 25%;  Two Hourly Exams: 20% each;  Final Exam: 35%.

Computer use:  Some of the Tuesday class sessions will take place in the computer lab in Weber 205. The dates in question will be announced in advance in the lecture.

Content:

Part I: First Order Equations

• Chapter 1: Introduction to Differential Equations. 1.1, 1.2, 1.3

• Chapter 2: First-order Equations. Solution techniques for linear and separable equations, exact equations, models of motion, autonomous equations and stability of equilibrium solutions. 2.1, 2.2, 2.3, 2.4, 2.6, 2.7, 2.9

• Chapter 3: Modeling and Applications. Personal finance. 3.3

• Chapter 6: Numerical Methods. Euler method. 6.1

Part II: Linear Second  Order Equations

• Chapter 4: Second-Order Equations. Homogenous and inhomogenous equations, variation of parameters and undetermined coefficients methods, forced and unforced harmonic motion. 4.2, 4.3, 4.4, 4.5, 4.6, 4.7

• Chapter 5: The Laplace Transform. Definition and properties, application to differential equations, discontinuous forcing terms, Delta function, convolution. 5.1, 5.2, 5.3, 5.4, 5.5, 5.6, 5.7

Part III: Systems of First Order Equations

• Chapter 7: Matrix Algebra. Vectors, matrices, linear systems of equations, subspaces, determinants - short review

• Chapter 8: Introduction to Systems. Definition, geometric interpretation, linear systems, phase-plane portraits. 8.1, 8.2, 8.3, 8.4, 8.5

• Chapter 9: Linear Systems with Constant Coefficients. Eigenvalue-eigenvector solutions of homogeneous systems and matrix exponential, phase-plane portraits and trace-determinant plane, qualitative analysis and stability, inhomogenous systems. 9.1, 9.5, 9.6 (9.2: planar systems), 9.3, 9.4, 9.7, 9.9

Part IV: Linear Higher  Order Equations

• Chapter 9.8: Higher Order Equations. Linear Dependence/Independence, Wronskian, fundamental set of solutions. 9.8, 4.3 revisited

Part V: Nonlinear Systems

• Chapter 10: Nonlinear Systems. Linearization, long-term behaviour of solutions, mechanical systems, population models.