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Appendix I
Cayley-Hamilton theorem
I.1
Statement of the theorem
According to the Cayley-Hamilton theorem, every square matrix A satises its
own characteristic equation (Volume 1, section 9.3.1). Let t
Computation of Harmonics
Let the Fourier series for the function y = f ( x ) is
f ( x) =
a0
+ an cos nx + + bn sin nx
2 n =1
n =1
where a0 =
1
2
f ( x ) dx ; a
n
=
0
1
(1)
2
f ( x ) cos nx dx ;
bn
SAT I
Assignment I
Differential and Difference Equations (MAT105)
1. Obtain the Fourier series for f ( x ) = e x in the interval 0 < x < 2 .
Hints: e ax cos bx dx =
f ( x ) = e x =
eax
e ax
ax
and
a
c
Fourier Series
1
Dirichlet conditions
The particular conditions that a function f (x) must
fulfil in order that it may be expanded as a Fourier
series are known as the Dirichlet conditions, and
may be
Submit on the Exercise Problems Monday (28-02-1913)
2 1 1
Problem 1: Find the characteristic equation of the matrix A = 0 1 0 , and hence compute
1 1 2
A1 . Also find the matrix represented by A8 5 A
Applications of Differential Equations - Second Order Equations
Series LCR Circuit
Consider a simple electrical circuit shown in the Figure, which consists of a resistor R in ohms; a capacitor C in
fa
Bessels Equation
Besselss equation is a second order differential equation of the form:
x2
dy
d2 y
+x
+ (x2 n2 )y = 0
dx2
dx
(1)
where n is a constant and is called the order of the Bessels equation.
SAT I
Assignment II
Differential and Difference Equations (MAT105)
1.
kx , 0 < x < l
Find the Fourier series expansion for f ( x ) =
repeating itself at intervals
0 , l < x < 2l
of 2l being k as cons
Chapter 1: Eigen Values and Eigen Vectors
Differential and Difference Equations
Prof. Suripeddi Srinivas
School of Advanced Sciences
VIT University
Vellore
[email protected]
Prof. S. Srinivas
M
Reading
[SB], Ch. 16.1-16.3, p. 375-393
1
Quadratic Forms
A quadratic function f : R R has the form f (x) = a x2 . Generalization
of this notion to two variables is the quadratic form
Q(x1 , x2 ) = a1
MAT105
UNIT 3 (Fourier series)
Fourier series: Let f (x) be a periodic (real-valued) function with period 2l satisfying the
following Dirichlet conditions on each interval I = [a, a + 2l ] IR1 whose l
Half-Range Series
23.5
Introduction
In this Section we address the following problem:
can we nd a Fourier series expansion of a function dened over a nite interval?
Of course we recognise that such a
UNIT V
www.rejinpaul.com
LAPLACE TRANSFORM
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VIT UNIVERSITY, VELLORE
School of advanced sciences
Winter Semester
Instructor: Dr. K. Raghavendar
Differential and Difference Equations (MAT105)
ASSIGNMENT 3
1. State the Dirichlets conditions for a
Differential and Difference (MAT 105)
Quiz I Winter 2013-14
Questions and Answers
[1]. State the condition for the Eigen values of a square matrix A and AT are same.
Ans: A I = AT I = 0
2 1 4
[2]. F
Practice problems in Fourier series
1. Obtain the Fourier series for f ( x ) = e x in the interval 0 < x < 2 .
Hints: e ax cos bx dx =
f ( x ) = e x =
e ax
e ax
ax
a
bx
+
b
bx
e
bx
dx
=
cos
sin
and
si
Chapter 1: Eigen Values and Eigen Vectors
Differential and Difference Equations
Prof. Suripeddi Srinivas
School of Advanced Sciences
VIT University
Vellore
[email protected]
Prof. S. Srinivas
M
Lecture notes
Dr.S.Srinivas
DDE /Slot F2
Senior Professor, SAS
_
Series solution of Differential Equations and Special functions
Validity of series solution of the equation:
d2y
dy
(i)
P0 ( x ) 2 + P1
VIT UNIVERSITY, VELLORE
School of advanced sciences
Winter Semester
Instructor: Dr. K. Raghavendar
Last date for Submission: 09-05-2014
Differential and Difference Equations (MAT105)
ASSIGNMENT IV
1.
VIT UNIVERSITY, VELLORE
School of advanced sciences
Winter Semester
Instructor: K. Raghavendar
Differential and Difference Equations (MAT105)
ASSIGNMENT-I
1. For the following matrices, find
(a) Eigen
Flash Card on Matrices, Vectors, Determinants, Linear System of Equations
Matrices
Denition. A matrix is a rectangular array of numbers (or functions) enclosed in brackets. These numbers (or
functions
For more Anna University Study Materials - search here - www.VidyarthiPlus.com/search.html
Unit. 1 Ordinary Differential Equations
UNIT I
ORDINARY DIFFERENTIAL EQUATIONS
Part A
The A.E is m2 m 1 0 m
Sturm Liouville Problems
Dr. S. Srinivas
February 18, 2014
1
Sturm-Liouville Differential Equation
The second order linear differential equation
d
dy
p(x)
+ [q(x) + r(x)]y = 0
dx
dx
for all a x b,
w
Legendre Equation
Dr. S. Srinivas
February 18, 2014
1
Solution of Legendre Equation
The Legendre equation of order is the second order linear differential equation:
(1 x2 )y 00 2xy 0 + ( + 1)y = 0,
(1
Difference Equations
Sequences often arise from recurrence relations. One type of recurrence relation is called a difference equation.
DEFINITION. A kth order linear difference equation (synonomously,
FACULTY OF MATHEMATICAL STUDIES
MATHEMATICS FOR PART I ENGINEERING
Lectures
MODULE 21
LAPLACE TRANSFORMS
1. Introduction
2. Transforms of simple functions
3. Properties of Laplace transforms
4. Invers
Chapter 1: Eigen Values and Eigen Vectors
Differential and Difference Equations
Prof. Suripeddi Srinivas
School of Advanced Sciences
VIT University
Vellore
[email protected]
Prof. S. Srinivas
M