Review of Power Series
Real Analytic functions
Ordinary points of a 2nd
Order Homogeneous
L.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 th
Particular Solutions of Non-Homogeneous
Linear Differential Equations with constant
coefficients
Method of Undetermined
Coefficients
In this lecture we discuss the Method of
undetermined Coefficients.
06/07/17
1
Consider the second order non-homogeneous
l
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 AND SCIENCE, PILANI
K K BIRLA GOA CAMPUS
DEPARTMENT OF MATHEMATICS
_
FIRST SEMESTER
(2012-2013)
OPTIMISATION
(AAOC C222)
TUTORIAL BOOKLET
Instructor-In-Charge
WA cA Wcfw_ttt
Table of Contents
Tutorial-1 .1
Tutorial-2 .4
Tuto
Solving linear Programming
Problems
Dr. Manoj Kumar Pandey
Problem
Wild West produces two types of cowboy hats. Type I
hat requires twice as much labor time as a Type II hat. If
all the available labor time is dedicated to Type II alone,
the company can p
TRANSPORTATION
PROBLEMS
Source
Destinations
c11 : x11
Units of supply
a1
1
1
b1
a2
2
2
b2
m
n
am
cmn : xmn
m - no. of sources
n - no. of destinations
cij transportation cost per unit
xij amount shipped
ai - amount of supply available at source i
bj amount
Non-linear Programming
Problem
Classical Optimization
Classical optimization theory uses differential
calculus to determine points of maxima and
minima for unconstrained and constrained
functions.
This chapter develops necessary and sufficient
condition
Goal programming
In this chapter we discuss the Goal programming
technique for solving multi-objective models (linear).
Goal programming problem is a problem of
finding solution which attains a predefined target
for one or more objective function.
If th
Integer Linear Programming
Integer Linear Programming
There are many LP problems in which the decision
variables can take only integer values.
If all the decision variables take only integer values it is
called a pure integer LPP; otherwise the problem is
1
Birla Institute of Technology and Science, Pilani-K. K. Birla Goa Campus
FIRST SEMESTER 2012-2013
Optimization
(AAOC C222 & MATH F212)
Tutorial Sheet-1
1. Taking the xj as variables, and all other symbols as given constants, determine whether
each of th
PERT Networks
In PERT the duration of any activity is
indeterministic. It bases the duration of an
activity on three estimates:
Optimistic Time, a
Most Likely Time, m
Pessimistic Time, b
The range [a, b] is assumed to enclose all
possible estimates of
Review of Power Series
Real Analytic functions
Ordinary points of a 2nd
Order Homogeneous
L.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 th
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) = L-1[F(p)].
1
1
1
L[1] L [ ] 1.
p
p
n!
n!
n
1
n
L[ x ]
L [
] x
n
1
n
1
p
p
n
1
x
L 1[
]
n!
p n1
17.12.16
2005-2006, Nadeem-ur-Rehm
Qualitative Properties of
solutions of a second order
homogeneous Linear
Differential equations.
Throughout this chapter we shall be
looking at the second order homogeneous
linear differential equation
y P ( x) y Q ( x ) y 0 .(1)
We shall like to say some
We assume a particular solution of
Ly k1e cos bx k2e sin bx
ax
ax
as y y p A1e cos bx A2e sin bx
ax
ax
We assume a particular solution of
ax
n
Ly e (b0 b1 x . bn x )
as y y p e ( A0 A1 x . An x )
We also use multiply by the least power of x
rule in case a
FROBENIUS SERIES
SOLUTION OF A SECOND
ORDER HOMOGENEOUS
LINEAR DIFFERENTIAL
EQUATION (C0NTINUED)
Assume that x = 0 is a regular singular point
of the second order homogeneous l.d.e.
y P ( x) y Q( x) y 0
Hence
p ( x) xP( x)
q( x) x Q( x)
2
are both analyti
Particular Solutions of Non-Homogeneous
Linear Differential Equations with constant
coefficients
Method of Undetermined
Coefficients
In this lecture we discuss the Method of
undetermined Coefficients.
12/17/16
1
Consider the second order non-homogeneous
l
Bessels Differential
equation (continued)
Dec 17, 2016
MATH C241 Prepared b
1
In this lecture we study properties of
Bessels functions, which are solutions of
Bessels equation.
Dec 17, 2016
MATH C241 Prepared b
2
Zeros of Bessel Functions
Fact:
(i) If 0 p
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
y-axis. The graph of odd functio
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
HOMOGENEOUS LINEAR
SYSTEM WITH CONSTANT
COEFFICENTS:
TWO EQUATIONS IN TWO
UNKNOWN FUNCTIONS
2005-2006, MATH C241 Prepared by Nadeem-ur-Rehman
12/17/16
1
We Shall concerned with the homogeneous
linear system
dx
a1 x b1 y
dt
dy
a2 x b2 y
dt
(1)
where th
LAPLACE TRANSFORMS
12/17/16
MATH C241
Prep
1
LAPLACE TRANSFORMS
Given a real-valued 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.
12/17/16
MATH C241
Prep
2
Thus the Lapl
Laplace Transform
17.12.16
2005-2006, Nadeem-ur-Rehman
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
EIGENVALUE PROBLEM
Dec 17, 2016
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
PARTIAL DIFFERENTIAL EQUATIONS:
THE VIBRATING STRING:
Suppose that a flexible string is pulled taut on
the x-axis and fastened at two points, which we
denote by x = 0 and x = L. The string is then
drawn aside into a certain curve y = f(x) in the
xy plane
Fourier Series
2005-2006, Nadeem-ur-Rehman
17.12.16
We will see that many important problems involving
partial differential equations can be solved,
provided a given function can be expressed as an
infinite sum of sines and cosines.
In this section and in
Bessels Differential
equation
Dec 17, 2016
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
Deterministic Dynamic Programming
Dynamic Programming (DP) determines the
optimum solution to an n-variable problem by
decomposing it into n stages with each stage
constituting a single-variable sub problem.
Recursive Nature of Computations in DP
Computat
Problemz 10 y y
Problem Set
Maximize = y1 2 n
10.3A Page 414
subject to y1+y2+yn = c,
yi 0
Thus there are n stages to this problem. At
stage i, we have to choose the variable yi.
The state of the problem at stage i is defined
by the variable xi, which rep