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 differe
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
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 equation
Bessels Differential
equation (continued)
Oct 21, 2017
MATH C241 Prepared b
1
In this lecture we study properties of
Bessels functions, which are solutions of
Bessels equation.
Oct 21, 2017
MATH C241
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
dra
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
Some Special Functions
of Mathematical Physics
Legendre Polynomials
(Continued)
Oct 21, 2017
1
In this lecture, we find the recurrence
relation satisfied by the Legendre
Polynomials, show that the nt
Applications to the
solutions of linear
differential equations
Oct 21, 2017
MATH C241 Prepar
1
Applications to the solutions of
linear differential equations
Consider the second order constant coeffic
EIGENVALUE PROBLEM
Oct 21, 2017
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 sol
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.
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 a
Laplace Transform
21.10.17
2005-2006, Nadeem-ur-R
ehman
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
equati
Bessels Differential
equation
Oct 21, 2017
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 B
HOMOGENEOUS LINEAR
SYSTEM WITH CONSTANT
COEFFICENTS:
TWO EQUATIONS IN TWO
UNKNOWN FUNCTIONS
10/21/17
2005-2006, MATH C241
Prepared by Nadeem-ur-R
1
We Shall concerned with the homogeneous
linear syste
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
Laplace Transform
21.10.17
2005-2006, Nadeem-ur-R
ehman
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
equati
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
dra
Sturm Liouville
Problems
Oct 21, 2017
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
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
LAPLACE TRANSFORMS
10/21/17
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
Q:
In a shipment of 15 room air conditioners, there are 5 with defective thermostats. Two air
conditioners will be randomly selected (without replacement) and inspected. What is the
probabilitythatthe
MVC Pattern
Model: classes for business logic and data model
View: UI components, displays data received from controller as result.
Can also transform the model(s) into UI
Controller: processes the
Packet Tracer Lab1
Introduction to Packet Tracer
What is Packet Tracer?
Packet Tracer is a protocol simulator developed by Cisco Systems. Packet Tracer is a powerful
and dynamic tool that displays the
Q: Prove that
1/2! + 2/3! + . + n/(n + 1)! = 1 -1/(n + 1)!, for all n in N .
Solution: We will prove this by mathematical induction.
LHS = 1/2! + 2/3! + . + n/(n + 1)!
RHS = 1 -1/(n + 1)!
So, taking f
Packet Tracker Lab2
Ethernet Broadcasts, Collisions Hubs and Switches
Part 1: Simple PDU and Hubs
Testing connectivity with PDUs
Once a topology has been created, connectivity can be tested between de
Q36
b)
(x)[P(x) Q(x)] (x)[P(x) V Q(x)] (x)[P(x) Q(x)]
Taking LHS as the expression to the left of the symbol , we will
try to deduce RHS
So,
LHS = (x)[P(x) Q(x)] (x)[P(x) V Q(x)]
RHS = (x)[P(x) Q(x)]
Problem 5: a) To find basis for Col(A), we need to reduce matrix A to row-echleon form.
Since, we can see that pivot of this matrix is in column 1 and 2, hence basis of this Col(A) = column 1 and
colu
Problem 6: Yes, there is a relationship between the rank of the homogeneous system of linear equations given by
A.x = 0 and the number of vectors in a basis for Col(A).
We first convert the homogeneou