Birla Institute of Technology & Science, Pilani  Hyderabad
MATHEMATIC 241

Spring 2015
BIRLA INSTITUTE OF TECHNOLOGY AND SCIENCE, PILANI (RAJASTHAN)
INSTRUCTION DIVISION
FIRST SEMESTER 20102011
Course Handout (Part II)
Aug 02, 2010
In addition to PartI (General Handout for all courses appended to the time table) this part gives
further sp
Birla Institute of Technology & Science, Pilani  Hyderabad
MATHEMATIC 241

Fall 2012
Bessels Differential
equation
Nov 25, 2014
MATH C241 prepared by MSR
1
In this lecture we discuss Bessels
Differential Equation. We also study
properties of Bessels functions, which are
solutions of Bessels equation. We first
review the definition and pro
Birla Institute of Technology & Science, Pilani  Hyderabad
MATHEMATIC 241

Fall 2012
HOMOGENEOUSLINEAR
SYSTEMWITHCONSTANT
COEFFICENTS
TWO EQUATIONS IN TWO
UNKNOWN FUNCTIONS
20052006, NadeemurRehman 1
We shall be concerned with the homogeneous
linear system
dx
=a1 x +b1 y
dt
dy
=a2 x +b2 y
dt
where the coefficients a1, a2, b1, and b2
Birla Institute of Technology & Science, Pilani  Hyderabad
MATHEMATIC 241

Fall 2012
IntroductiontoSystemsofFirst
OrderLinearEquations
20052006, NadeemurRehman
1
A system of simultaneous first
order
ordinary
differential
equations has the general form
y1 = f1 ( x, y1, y2 ,K yn )
y2 = f 2 ( x, y1, y2 ,K yn )
M
yn = f n ( x, y1, y2 ,K yn
Birla Institute of Technology & Science, Pilani  Hyderabad
MATHEMATIC 241

Fall 2012
Applicationstothesolutions
oflineardifferential
equations
Nov 25, 2014
MATH C241 Prepared by MSR
1
Applicationstothesolutionsoflinear
differentialequations
Consider the second order constant coefficient
linear differential equation
y +a y+b y = f ( x )
wi
Birla Institute of Technology & Science, Pilani  Hyderabad
MATHEMATIC 241

Fall 2012
SturmLiouvilleProblems
Nov 25, 2014
MATH C241 Prepared by MSR
1
Second Order Exact Differential equations
The second order linear differential equation
P ( x) y +Q ( x) y+R ( x) y =0
is called exact if it can be written in the form
( P( x) y)+( S ( x) y )
Birla Institute of Technology & Science, Pilani  Hyderabad
MATHEMATIC 241

Fall 2012
Inverse Laplace Transform:
If L[f(x)] = F(p), then f(x) is called an inverse
Laplace transform of F(p), and we write
f(x) = L1[F(p)].
1
1
1[ ] 1.
L[1] L
p
p
n ] n! L1[ n! ] x n
L[ x
n
1
n
1
p
p
1
xn
L1[
]
1
n!
p n
25.11.14
20052006, NadeemurRehman
1
Birla Institute of Technology & Science, Pilani  Hyderabad
MATHEMATIC 241

Fall 2012
LAPLACETRANSFORMS
11/25/14
MATH C241
Prepared by MSR
1
LAPLACETRANSFORMS
Given a realvalued function f(x) defined for
all x 0, we define its Laplace transform as
L[ f ( x)]
e
px
f ( x)dx F ( p )
0
where p is a real number.
11/25/14
MATH C241
Prepared b
Birla Institute of Technology & Science, Pilani  Hyderabad
MATHEMATIC 241

Fall 2012
ReviewofPowerSeries
RealAnalyticfunctions
Ordinarypointsofa2ndOrder
HomogeneousL.D.E.
In this lecture we discuss the Power series
method of solving a second order
homogeneous linear differential equation
First we spend some time in reviewing the
results a
Birla Institute of Technology & Science, Pilani  Hyderabad
MATHEMATIC 241

Fall 2012
Applicationstothesolutions
oflineardifferential
equations
Nov 25, 2014
MATH C241 Prepared by MSR
1
Applicationstothesolutionsoflinear
differentialequations
Consider the second order constant coefficient
linear differential equation
y +a y+b y = f ( x )
wi
Birla Institute of Technology & Science, Pilani  Hyderabad
MATHEMATIC 241

Fall 2012
EiGENVALUE PROBLEM
Nov 25, 2014
MATH C241 Prepared by MSR
1
In this lecture we study Boundary value
problems for Ordinary Differential
equations. These problems arise in Physics
when we attempt to solve the problem of
heat conduction, vibrating strings et
Birla Institute of Technology & Science, Pilani  Hyderabad
MATHEMATIC 241

Fall 2012
Bessels Differential
equation (continued)
Nov 25, 2014
MATH C241 Prepared by MSR
1
In this lecture we study properties of
Bessels functions, which are solutions of
Bessels equation.
Nov 25, 2014
MATH C241 Prepared by MSR
2
Zeros of Bessel Functions
Fact:
Birla Institute of Technology & Science, Pilani  Hyderabad
MATHEMATIC 241

Spring 2015
1
First A. Author Bhawika Jain, Member, Mathematics Department
A mathematical model for Price War
Abstract In this paper we present a nonlinear
mathematical model based on system of differential equation
for Price War Here three variables, namely the num
Birla Institute of Technology & Science, Pilani  Hyderabad
MATHEMATIC 241

Spring 2015
BIRLA INSTITUTE OF TECHNOLOGY AND SCIENCE, Pilani
Pilani Campus
Instruction Division
SECOND SEMESTER 20132014
Course Handout Part II
In addition to partI (General Handout for all courses appended to the time table) this portion gives further
specific de
Birla Institute of Technology & Science, Pilani  Hyderabad
MATHEMATIC 241

Spring 2015
i
TEST
IQ
YOUR
ii
This page intentionally left blank
TEST
IQ
400 questions to
boost your brainpower
Philip Carter & Ken Russell
YOUR
iv
First published in 2000 Reprinted in 2001, 2004 Reissued in 2007 Apart from any fair dealing for the purposes of resear
Birla Institute of Technology & Science, Pilani  Hyderabad
MATHEMATIC 241

Spring 2015
i
IQ AND
PERSONALITY
TESTS
ii
This page is left intentionally blank
iii
IQ AND
PERSONALITY
TESTS
PHILIP CARTER
London and Philadelphia
iv
Publishers note
Every possible effort has been made to ensure that the information contained in this
book is accurate
Birla Institute of Technology & Science, Pilani  Hyderabad
MATHEMATIC 241

Fall 2012
SomeSpecialFunctionsof
MathematicalPhysics
LegendrePolynomials
21 SEP 2005
1
There are many special functions that are
being used by Physicists in the solution of
physical problems, like, Legendre
Polynomials, Hermite Polynomials, Bessels
Functions, etc.
Birla Institute of Technology & Science, Pilani  Hyderabad
MATHEMATIC 241

Fall 2012
Laplace Transform
25.11.14
20052006, NadeemurRehman
1
In this chapter we will introduce a technique
as the Laplace transform, which is a very
useful tool in the study of linear differential
equations. Although be no means limited to
this class of probl
Birla Institute of Technology & Science, Pilani  Hyderabad
MATHEMATIC 241

Fall 2012
Second Order Linear Differential equations
In this lecture
We define a second order Linear DE
State the existence and uniqueness of solutions
of second order Initial Value problems
We find the general solution of a second order
Homogeneous Linear DE
D
Birla Institute of Technology & Science, Pilani  Hyderabad
MATHEMATIC 241

Fall 2012
Second Order Constant Coefficient
Homogeneous Linear Differential
equations
In this lecture
We give methods for finding the general
solution of a second order homogeneous
linear differential equations with constant
coefficients.
A second order homogeneou
Birla Institute of Technology & Science, Pilani  Hyderabad
MATHEMATIC 241

Fall 2012
LinearDifferentialEquations
In this lecture we discuss the methods
of solving first order linear differential
equations and reduction of order.
LinearEquations
Alinearfirstorderequationisanequation
thatcanbeexpressedintheform
dy
a1 ( x) +a0 ( x) y =b( x),
Birla Institute of Technology & Science, Pilani  Hyderabad
MATHEMATIC 241

Fall 2012
DifferentialEquations
MATHC241
Class hours: T Th S 2
(9.00 A.M. to 9.50 A.M.)
TextBook:DifferentialEquations
withApplicationsand
HistoricalNotes:
by George F. Simmons
(Tata McGrawHill) (2003)
In this introductory lecture, we
Define a differential equati
Birla Institute of Technology & Science, Pilani  Hyderabad
MATHEMATIC 241

Fall 2012
GAUSSS
HYPERGEOMETRIC
EQUATION(Continued)
Nov 25, 2014
MATH C241 Prepared by MSR
1
Abstract
In this lecture we continue the study of
Hypergeometric equation. We also show
that certain second order homgeneous l.d.e.
can be transformed into a hypergeometric
Birla Institute of Technology & Science, Pilani  Hyderabad
MATHEMATIC 241

Fall 2012
PARTIALDIFFERENTIAL
EQUATIONSAND
BOUNDARYVALUE
PROBLEMS
VIBRATINGSTRINGSAND
HEATCONDUCTION
11/25/14
MATH C241
Prepared by MSR1
THE VIBRATING STRING:
Suppose that a flexible string is pulled taut on
the xaxis and fastened at two points, which
we denote by
Birla Institute of Technology & Science, Pilani  Hyderabad
MATHEMATIC 241

Fall 2012
Even and Odd Functions
A function f, defined on an interval centred at
the origin is said to be even if
f ( x) f ( x)
For all x in the domain of f, and odd if
f ( x) f ( x)
The graph of even function is symmetric about the
yaxis. The graph of odd functio