Differential Equations
Instructors: Dr. Dariusz Bugajewski (Lecture)
Consultant: ISU faculty member Dr. Fritz Keinert
This course is based on the ISU course MATH267 (4 credits).
The corresponding NCSU course is MA341 (4 credits).
Text: Zill,
Differential Equations with Boundary Value Problem s Cengage/9E Edition ISBN:
9781337604901
The course is based on MA267 at ISU. At ISU
there is also an option for MA266, which is the same except for two additional
topics, Laplace Transforms and Power Series Methods that are in MA 267. We will
offer all of the topics, and therefore only MA267. MA267 includes solution
methods for ordinary differential equations. First order equations, linear
equations, constant coefficient equations. Eigenvalue methods for systems of
first order linear equations. Introduction to stability and phase plane
analysis. It is related to the course MA341 taught at NCSU is, if anything, a
slightly more difficult course. Since Prof. Bugajewski
has taught the NCSU variant he can take into consideration what students from
NCSU will need to know. Dr. Bugajewski has done
proofs that were additional to justify the material. Students receive a firm
foundation in this class that is suitable for engineering programs as well as
chemistry and physics.
Course Objectives
Be able to identify types of differential
equations and use appropriate methods to solve them.
Be able to use the method of integrating
factors to solve first order linear equations. Be able to separate variables
and compute integrals in solving first order separable equations. Know how to
find a general solution of a linear second order constant coefficient
homogeneous differential equation by seeking exponential solutions. Be able to
use the method of undetermined coefficients to find a particular solution of a
linear second order constant coefficient nonhomogeneous differential equation.
Be able to find a general solution of a linear second order constant
coefficient nonhomogeneous equation. Be able to solve an initial value problem
associated with a linear second order constant coefficient homogeneous or
nonhomogeneous equation. Be able to extend the methods used for linear second
order constant coefficient equations to higher order linear constant
coefficient equations, both homogeneous and non-homogeneous. Be able to use the
eigenvalue-eigenvector method to find general solutions of linear first order
constant coefficient systems of differential equations of size 2 or 3. Be able
to find a fundamental matrix for linear first order constant coefficient system
of differential equations of size 2 or 3. Be able to use the method of
variation of parameters to find a particular solution of a nonhomogeneous
linear first order constant coefficient system of size 2.
Learn how differential equations are used to model
physical systems and applied problems.
Be able to formulate and use elementary models
for population dynamics, such as the logistic equation, to describe transient
and steady state behavior. Be able to work with models for the linear motion of
objects using assumptions on the velocity and acceleration of the object. Be
able to set up and solve a problem involving stirred tank reactor dynamics. Be
able to use Newton’s second law to set up a model for a simple spring-mass
system; and use appropriate methods to obtain the solution of the model
problem. Be able to use models for continuous compounding of interest to
describe elementary savings and loan problems.
Gain an elementary understanding of the theory
of ordinary differential equations.
Understand statements on existence and
uniqueness of solutions. Understand the role of linear independence of
solutions in finding general solutions of differential equations. Understand what
constitutes a general solution of a differential equation.
Additional objectives for MA267
Be able to use the method of Laplace transforms
to solve linear second order constant coefficient homogeneous and
nonhomogeneous equations. Be able to use series methods to find a power series
solution of a linear second order variable coefficient homogeneous equation
about an ordinary point.
Course outline
Chapter & Sections Topics
Chapter 1 – Introduction §§1.1-3
Chapter 2 - First Order ODEs §§2.1-5 §2.6
(optional)
Chapter 3 - First Order ODE Models selected
topics from §§3.1-2 §3.3 (optional)
Chapter 4 - Higher Order Linear ODEs §§4.1-4,
6, 7, 9
Chapter 5 - Higher Order ODE Models selected
topics from §5.1
Chapter 8 - First Order Systems §§8.1-2 §8.3
(optional)
Chapter 9 - Numerical Solutions of ODEs §9.1
(optional)
Chapter 10 - Planar Autonomous Systems §10.1
selected topics from §§10.2-3 (optional)
Math 267 only
Chapter 7 - Laplace Transform §§7.1-5
Chapter 6 - Power Series Methods §§6.1-2