Introduction to Ordinary Differential Equations

**Instructor **Ass.
Professor Iuliana Oprea, Weber 123

**Office Hours: **MW
9:10-10:00AM, and by appt., Phone:
491-6751 Office

Email:
juliana"at"math.colostate.edu; www: http://www.math.colostate.edu/~juliana/M340.html

**Class Time and Room: **MTWF
8:00-8:50AM in EE 203

Lab Section: This
course is formally split in a lecture and a lab session, which in
practice will not be separated. You should register, however, for both
lecture (338813) and Lab (338814).

__Required Textbook__:

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

Supplementary Text:

J. Polking, D. Arnold: Ordinary Differential Equations using Matlab
(available shrink-wrapped with the textbook at no extra costs)

** 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. Each
homework you hand in should have a header at the top of the first page
with your name, the date you hand in the homework, and the number of
the
homework (e.g. Homework 1 etc).

**Examinations:** There will be two in-class exams: February 22 (Wed), April 12 (Wed),
and a Final Exam on May 9( Tues)
1:30-3:30p

**Grading:** Graded Homework,
quizzes: 25%; Two Hourly Exams: 20%
each; Final Exam: 35%. Grades will
be published on WebCT.

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.

Tutoring: free tutoring for M340 is available at http://www.colostate.edu/Depts/NatSci/html/Tutorial.html

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**

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

**• Chapter 7: Matrix Algebra. **Vectors,
matrices, linear systems of equations, subspaces, determinants. **7.1,
7.2, 7.3, 7.4, 7.5, 7.6, 7.7**

**• 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
**

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

**Part V: Linear Higher
Order Equations**

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