Chapter 8
Series Solutions of
Differential Equations
p.451
8.3
8.4
8.5
8.6
8.7
8.1
The Taylor Polynomial
Approximation
p.451
Dr. A.Didenko.
1
Taylor series of the function f at a
(or about a or centered at a).
(3) f(x)
an x a
j
j0
f(a) f (a) x-a
f (j)(
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Assignment: Individual Literature Review Preparation Draft 1
Introduction:
Write a paragraph explaining the significance of your chosen topic in general and your particular
interest in this topic. State the main objective(s) of yo
Name:
ID:
Section:
Date:
Assignment: Individual Literature Review Preparation Draft 1
Introduction:
Write a paragraph explaining the significance of your chosen topic in general and your particular
interest in this topic. State the main objective(s) of yo
Fourier Series
Lesson 26
Fourier Series
1
Objectives:
At the end of the lesson the student should
be able to:
Define periodic function.
Determine a period of a function.
Find a Fourier Series of a given function.
Fourier Series
2
Introduction
Periodicp
Fourier Series
Lesson 26
Fourier Series
1
Outcomes
At the end of the lesson the student should
be able to:
Define periodic function.
Determine a period of a function.
Find a Fourier Series of a given function.
Fourier Series
2
Introduction
Periodicphen
First Order Differential Equations
Separable Variables
L2.1
1
Learning Outcomes
The students should be able to :
Solve a first order differential equation
by using Separable Variables.
2
Simple Recall!
Integrate the following :
1.
3.
3
(3x 5 x
1
)dx
2 x
Exact Equations
C h a p te r2
Le s s o n6
UTP/JBJ
1
Objectives
Attheendofthelessonthestudentsshouldbe
ableto:
1.Defineanexactdifferential
equation.
2.Solveanexactdifferential
equation
UTP/JBJ
2
ExactEquation
Definition:
Afirstorderdifferentialequationof
Differential Equations
Lesson 1
1
Introduction
Definition
Equations containing derivatives are
differential equations.
Derivatives are rates of change i.e.,
-motion of fluids
-flow of current in electric circuits
-dissipation heat from solid objects
-prop
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Helps the team write its background/literature
review in the proposal and the final research
report.
Each member selects two texts based on
4.4
Nonhomogeneous
Equations:
The Method of
Undetermined
Coefficients
4.5
4.6
p.188
4.7
4.9
Example 1, p.188
2
Dr. A. Didenko
1
Example 2, p.189
3
Example 3, p.189
4
Dr. A. Didenko
2
Example 4, p.190
5
Example (9), p.191
y + y = 5
6
Dr. A. Didenko
3
Examp
Chapter 1
Introduction
1.1
Background
p.1
Dr. A.Didenko.
1
Example
differential equations, order of a differential equation
a ) f ( x) cos x, f ( x)
b) y( x) y ( x) 3 x 2 x 3 , y1 ( x)
y2 ( x) e x x 3
c) y( x) y ( x) 0,
y1 ( x)
, y2 ( x) cos x
3
What
Chapter 4
Linear Second-Order
Equations
p.163
4.1
Mass-Spring
Oscillator
p.163
Dr. A.Didenko
1
kk
m
m - inertia
b
y
Equilibrium
point
Hookes law
Damped oscillator
Fspring = -ky
Ffriction = -b(dy/dt) = -by
stiffness = k > 0
damping coefficient = b 0
4.1
3
Partnering to Further Enhance Pipeline Safety
In Communities
Through Risk-Informed Land Use Planning
Final Report of Recommended Practices
November 2010
The Pipelines and Informed Planning Alliance is sponsored by the United States Department of
Transport
Blowout:TheDeepwaterHorizonDisaster
A Survivor Recalls His Harrowing Escape but who is ultimately to blame?
May 10, 2010 CBS News
The Deepwater Horizon drilling rig was owned and operated by the company Transocean and
was leased to client oil company BP.
Criteria For
Source Evaluation
Assessing the quality and usefulness of an article
To help you make a decision about the quality
and usefulness of any article or book, you
need to ask questions about seven factors:
1. Author
2. Content
3. Audience
4. Reada
INTRODUCTION TO
ENGINEERING DESIGN
ENGR110
The Design Process
2
Design versus other aspects of engineering
Overview of the process
Steps in the design process
ENGR 110 - Introduction to Engineering
Engineering Functions
3
Design
Manufacture - assemble com
Paraphrasing
Agenda
1. Definition
2. Purpose
3. Parts of a paraphrase
4. Recognizing a correct paraphrase
5. Creating a correct paraphrase
Definition
A paraphrase is a point by point explanation of
another persons ideas written in your own
words.
(Spatt,
Name _
Student ID _
SCORE
/14
ENGR 110 Syllabus Assignment
A course syllabus is a guide for student success. It is a document that tells you where you will
end up when the semester is over, and how you will get there. Most instructors go over the
course s
CHAPTER2
Section3
LinearEquations.
1
08/09/12
Azizan : Linear Equations
Definition
AfirstorderDEoftheform
dy
a1 ( x )
+ a0 ( x ) y = g ( x )
dx
issaidtobealinearequation.
2
Azizan : Linear Equations
08/09/12
g ( x) = 0
Whenthelinearequationissaidto
behomo
DIFFERENTIAL EQUATIONS
CHAPTER 1
S e c tio n 2
S o lutio ns o f an ODE
08/09/12
Azizan:Solution of an ODE
1
Definition: A s o lutio no f
(
)
F x, (x) , (x),. . , (x) = 0
(n)
isafunctionthathasatleast
n
08/09/12
derivatives.
Azizan:Solution of an ODE 2
y=
CHAPTER 7 Section 2
LAPLACE TRANSFORM
08/09/12
1
SECTION1
DEFINITION
08/09/12
2
For
t 0, f ( t )
is a function
st
Lcfw_ f (t ) = e f (t )dt
0
provided the integral converges, i.e.
the limit
Note:
08/09/12
b st
lim e dt
b 0
exists!
Lcfw_ f (t ) = F ( s )
CHAPTER 7
Section 3
Translation on the saxis.
08/09/12
Azizan:Translationonthesaxis.
1
Using the definition to find Laplace
transform of a function is often
inconvenient. It is now possible to compute
Lcfw_ e f ( t ) by translating or shifting.
at
08/09/
Chapter 7
Section 3, part ii.
Translation on the t-axis.
08/09/12
Azizan: Translation on the t-axis
1
Unit Step Function
08/09/12
Azizan: Translation on the t-axis
2
Definition: The unit step function
U ( t a)
is defined to be
0 , 0 t < a
U ( t a) =
ta
1