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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 the characteristic
polynomial of A be
() = det [A 1].
(I
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 =
0
1
2
f ( x ) sin nx dx
(2)
0
Let the function is gi
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
cos
bx
+
b
sin
bx
e
sin
bx
dx
=
(
)
( a sin bx b cos bx
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 summarized by the following points:
1. the function mu
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 A7 + 7 A6 3 A5 + A4 5 A3 + 8 A2 2 A + I .
3 1 1
Proble
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
farads; an inductor L in henries; and an electromotive fo
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. This equation can be written in the the
(Sturm-Liouvill
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 constant.
0
kl
Hints: a0 = , an = 2kl
2
n 2 2
2.
when n is
Chapter 1: Eigen Values and Eigen Vectors
Differential and Difference Equations
Prof. Suripeddi Srinivas
School of Advanced Sciences
VIT University
Vellore
srinusuripeddi@gmail.com
Prof. S. Srinivas
MAT 105 DDE
srinusuripeddi@gmail.com
1 / 10
Chapter 1: E
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 ) = a11 x21 + a12 x1 x2 + a21 x2 x1 + a22 x22 .
Here each ter
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 length equals the
period 2l :
(a) f (x) has only a finit
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 function could not be periodic (as periodicity demands
UNIT V
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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 function to have a Fourier series expansion.
2. Does th
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]. Find the Eigen values of the matrix A = 0 3 2 .
0 0 6
A
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
sin
(
)
( a sin bx b cos bx )
a 2 + b2
a 2 + b2
1 e 2 1 1
Chapter 1: Eigen Values and Eigen Vectors
Differential and Difference Equations
Prof. Suripeddi Srinivas
School of Advanced Sciences
VIT University
Vellore
srinusuripeddi@gmail.com
Prof. S. Srinivas
MAT 105 DDE
srinusuripeddi@gmail.com
1 / 17
Chapter 1: E
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 ( x ) + P2 ( x ) y = 0
dx
dx
can be determined with th
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. Form the difference equation by eliminating the arbitra
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 values, Eigen vectors
(b) Verify Cayley-Hamilton theor
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) are called entries or elements of the matrix.
Example
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
om
Problem 1 Solve the equation D 2 D 1 y 0
Solution:
1
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,
where p. p0 , q and r are continuous on [a, b] and p(x)
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,
(1.1)
where the parameter is a real number greater than 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, a linear recurrence relation) is a set of equations of
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. Inverse Laplace transform
5. Solution of differential equatio
Chapter 1: Eigen Values and Eigen Vectors
Differential and Difference Equations
Prof. Suripeddi Srinivas
School of Advanced Sciences
VIT University
Vellore
srinusuripeddi@gmail.com
Prof. S. Srinivas
MAT 105 DDE
srinusuripeddi@gmail.com
1 / 21
Chapter 1: E