LECTURE # 18
Article 29
Gauss Hyper geometric Equation
Gauss Hyper geometric Equation
The equation is :
x(1 x ) y + [c (a + b + 1)x] y aby = 0
where a, b, c are constants.
(1)
Here ,
c (a + b + 1)x
ab
P(x ) =
, Q (x ) =
x(1 x )
x(1 x )
x = 0 & x = 1 are
Lecture #14
Series Solution of First O.D.E
Section 27
Series Solution of First Order D.E.
(1)In each of following D.E. find a power series
an x n , try to recognize the
solution of the form
resulting series as the expansion of a familiar function,
and ve
Section 26
Review of Power series
Lecture # 13
Review of Power Series
An infinite series of the form
an x n = a0 + a1 x + a2 x 2 + .
(1)
n =0
is called a power series in x.
The series
a (x x )
n
n
0
= a0 + a1 ( x x0 ) + a2 (x x0 ) + .
2
n=0
is a power se
LECTURE #12
The STURM Comparison Theorem
Section 25
The STURM Comparison Theorem
Theorem A :
Let
y(x )
be a non-trivial solution of
y + q( x ) y = 0
(1)
on a closed interval [a,b]. Then
y(x )
has
at most a finite number of zeros in this interval.
Theorem
LECTURE # 10
Section 23
Operator methods for finding
particular solution
Section 23
Operator methods for finding particular solution
(a)
D
dy
d2y
dny
Dy = , D 2 y = 2 ,., D n y = n
dx
dx
dx
differential operator.
1
(b) If y =
f (x )
D
1
f ( x ) = f ( x )d
Lecture # 9
The method of variation of parameters
Section 19
The method of variation of parameters
Let
y + P( x ) y + Q(x ) y = R( x ) (1)
where
P(x ), Q( x ), R( x )
are function of x or const.
Let the homogeneous equation of (1) be
y + P( x ) y + Q( x )
Lecture # 8
The Method of Undetermined Coefficients
Section 18
The Method of Undetermined Coefficients
Qn. 3)
If
y1 ( x) and y2 ( x) are solution of
y '+ P( x) y '+ Q( x) y = R1 ( x)
and y '+ P( x) y '+ Q( x) y = R2 ( x)
then y( x) = y1 ( x)+ y2 ( x) is a
LECTURE # 7
LECTURE
The use of a known solution to find
another
another
The Homogeneous Equation with
constant coefficient
constant
ARTICLE 16
ARTICLE
The use of known solution to find
The
another
another
Let y + P(x)y + Q(x)y = 0
Let
be a homogeneous equ
LECTURE #5,#6
Section 14-15
Introduction
The general solution of a Homogeneous
Equation
A second order linear diff eqn. (lde) is eqn. of the
second
of
form
form
a0(x)y + a1(x)y + a2(x)y = h(x)
where a0(x), a1(x), a2(x), h(x) are continuous
(x),
functions
LECTURE # 4
Section 10 &11
Linear Equations
Reduction of Order
Section 10
Linear Equations
To solve the equation
dy
= 2 xy ( y > 0 )
dx
(1)
we multiply both sides by the factor 1/y to
get,
1 dy
.
= 2 x; that is, Dx (ln y ) = Dx x 2
y dx
()
(2)
Because eac
LECTURE # III
Section 09
Integrating Factors
Section 09
Integrating Factors
Exact differential equations are comparatively rare,
for, exactness depend on a precise balance in the
form of the equation and is easily destroyed by
minor changes in the form.
C
LECTURE # II
Section 07 -08
Separable Equations and applications
Implicit solutions and General Solutions
Homogeneous Equations
Equations reducible to homogeneous form
Exact equations
Linear vs. Nonlinear Differential
Equations
An ordinary differential eq
Differential Equations
MATH C241
Text Book: Differential Equations with
Applications and Historical Notes: George
F. Simmons, 2ed.,TMH
Lecture Notes prepared by:
Dr. Bimal Kumar Mishra
Mathematics Group
LECTURE # I
Topics to be covered:
Section 01 - 06
Wh