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
HOMOGENEOUS LINEAR
SYSTEM WITH CONSTANT
COEFFICENTS:
TWO EQUATIONS IN TWO
UNKNOWN FUNCTIONS
08/09/15
2005-2006, MATH C241
Prepared by Nadeem-ur
1
We Shall concerned with the homogeneous
linear system
dx
a1 x b1 y
dt
dy
a2 x b2 y
dt
(1)
where the coeff
Sturm Liouville
Problems
Aug 9, 2015
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 )
IntroductiontoSystemsofFirst
OrderLinearEquations
2005-2006, Nadeem-ur-R
ehman
1
A system of simultaneous first
order ordinary differential
equations has the general form
x1 F1 (t , x1, x2 ,K xn )
x2 F2 (t , x1, x2 ,K xn )
M
xn Fn (t , x1, x2 ,K xn )
2
GAUSSS
HYPERGEOMETRIC
EQUATION (Continued)
Aug 9, 2015
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
Some Special Functions
of Mathematical Physics
Legendre Polynomials
(Continued)
Aug 9, 2015
1
In this lecture, we find the recurrence
relation satisfied by the Legendre
Polynomials, show that the nth degree
Legendre Polynomial has n distinct real
roots a
Some Special Functions
of Mathematical Physics
Legendre Polynomials
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,
EIGENVALUE PROBLEM
Aug 9, 2015
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 etc
GAUSSS
HYPERGEOMETRIC
EQUATION
Aug 9, 2015
MATH C241 Prepared by MSR
1
Abstract
In this lecture we discuss the famous 2nd
order homogeneous l.d.e. the so called
Gausss hypergeometric d.e. We also show
that many elementary functions are
solutions of suitab
Bessels Differential
equation (continued)
Aug 9, 2015
MATH C241 Prepared b
1
In this lecture we study properties of
Bessels functions, which are solutions of
Bessels equation.
Aug 9, 2015
MATH C241 Prepared b
2
Zeros of Bessel Functions
Fact:
(i) If 0 p <
LAPLACE TRANSFORMS
08/09/15
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.
08/09/15
MATH C241
Prep
2
Thus the Lap
Bessels Differential
equation
Aug 9, 2015
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 prop
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
2005-2006, Nadeem-ur-Rehman
1
A system of simultaneous first
order ordinary differential
equations has the general
form
x1 F1 (t , x1, x2 ,K xn )
x2 F2 (t , x1, x2 ,K xn )
M
xn Fn (t , x1, x2 ,K xn )
2005-2006, Nadeem-ur-Rehman
(1)
2
where each xk is a
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[ ] 1.
L[1] L
p
p
n ] n! L 1[ n! ] x n
L[ x
n1
n1
p
p
1
xn
L 1[
]
n!
p n1
09.08.15
2005-2006, Nadeem-ur-Rehman
1
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
Fourier Series
2005-2006, Nadeem-ur-Rehman
09.08.15
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
Laplace Transform
09.08.15
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
HOMOGENEOUS LINEAR
SYSTEM WITH CONSTANT
COEFFICENTS:
TWO EQUATIONS IN TWO
UNKNOWN FUNCTIONS
2005-2006, MATH C241 Prepared by Nadeem-ur-Rehman
08/09/15
1
We Shall concerned with the homogeneous
linear system
dx
a1 x b1 y
dt
dy
a2 x b2 y
dt
(1)
where th
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
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
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
FIRST ORDER
DIFFERENTIAL EQUATIONS
In this lecture we discuss various methods
of solving first order differential equations.
These include:
Variables separable
Homogeneous equations
Exact equations
Equations that can be made exact by
multiplying by an
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.
08/09/15
1
Consider the second order non-homogeneous
l
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
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
Linear Differential Equations
In this lecture we discuss the methods
of solving first order linear differential
equations and reduction of order.
Linear Equations
A linear first order equation is an equation that
can be expressed in the form
dy
a1 ( x) a0
Differential Equations
MATH C241
Class hours: T Th S 2
(9.00 A.M. to 9.50 A.M.)
Text Book:
Differential Equations
with Applications and
Historical Notes:
by George F. Simmons
(Tata McGraw-Hill) (2003)
In this introductory lecture, we
Define a differentia