International Journal of Trend in Scientific Research and Development (IJTSRD)
Volume 3 Issue 6, October 2019 Available Online: www.ijtsrd.com e-ISSN:
e
2456 – 6470
The Super Stability off Third Order Linear Ordinary Differential
Homogeneous Equation with
ith Boundary Condition
Dawit Kechine Menbiko
Department off Mathematics, College of Natural and Computational Science, Wachemo University, Ethiopia
How to cite this paper:
paper Dawit Kechine
Menbiko "The Super Stability of Third
Order Linear Ordinary Differential
Homogeneous Equation with Boundary
Condition" Published
in
International
Journal of Trend in
Scientific Research
and
Development
(ijtsrd), ISSN: 24562456
IJTSRD29149
6470, Volume-3
Volume
|
Issue-6,
6,
October
2019,
pp.697
pp.697-709,
URL:
https://www.ijtsrd.com/papers/ijtsrd29
149.pdf
ABSTRACT
The stability problem is a fundamental issue in the design of any distributed
systems like local area networks, multiprocessor systems, distribution
computation and multidimensional queuing systems. In Mathematics
stability theory addresses the stability
tability solutions of differential, integral and
other equations, and trajectories of dynamical systems under small
perturbations of initial conditions. Differential equations describe many
mathematical models of a great interest in Economics, Control theory,
theo
Engineering, Biology, Physics and to many areas of interest.
In this study the recent work of Jinghao Huang, Qusuay. H. Alqifiary, and
Yongjin Li in establishing the super stability of differential equation of
second order with boundary condition was extended to establish the super
stability of differential equation third order with boundary condition.
KEYWORDS: supper stability, boundary conditions, Intial conditions
INTRODUCTION
In recent years, a great deal of work has been done on
various aspects of differential equations of third order.
Third order differential equations describe many
mathematical models of great iterest in engineering,
biology and physics. Equation of the form
x′′′ + a(x) x′′ + b(x)x′ + c(x)x = f (t) arise in the study of
entry-flow phenomena, a problem of hydrodynamics
which is of considerable importance in many branches of
engineering.
There are different problems concerning third order
differential equations which have drawn the attention of
researchers throughout the world. [17]
In mathematics stability theory addresses the stability
st
of
solutions of differential equations, Integral equations,
including other equations and trajectories of dynamical
systems under small perturbations. Following this,
stability means that the trajectories do not change too
much under small perturbations [7].The
The stability problem
is a fundamental issue in the design of any distributed
systems like local area networks, multiprocessor systems,
mega computations and multidimensional queuing
systems and others. In the field of economics, stability is
achieved
ieved by avoiding or limiting fluctuations in
production, employment and price.
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y/4.0)
The stability problem in mathematics started by Poland
mathematician Stan Ulam for functional equations around
1940; and the partial solution of Hyers to the Ulam’s
problem [2] and [20].
In 1940, Ulam [21] posed a problem concerning the
stability of functional equations:
“Give conditions in order for a linear function near an
approximately linear function to exist.”
A year later, Hyers answered to the problem of Ulam for
additive functions defined on Banach space: let and be real Banach spaces and ɛ 0. Then for every function
f : X1 → X 2
Satisfying
f ( x + y ) − f ( x) − f ( y ) ≤ ε
x, y ∈ X 1
There exists a unique additive function
A : X1 → X 2
with the property
f ( x) − A( x) ≤ ε
x ∈ X1
Thereafter, Rassias [14] attempted to solve the stability
problem of the Cauchy additive functional equations in
more general setting. A generalization of Ulam’s problem
is recently proposed by replacing functional equations
with differential equations
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ϕ ( f , y, y′,... y ( n ) ) = 0
September - October 2019
and
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has the Hyers-Ulam stability if for a given
function y such that
ε >0
and a
ϕ ( f , y, y′,... y ( n ) ) ≤ ε
There exists a solution
Main result
Super stability with boundary condition
Definition: Assume that for any function
y satisfies the differential inequalities
y0 of the differential equation
such that
y ∈ c n [a, b] ,if
ϕ ( f , y, y ',..., y ( n ) ) ≤ ε K.....(1)
x ∈ [ a, b] and for some ε ≥ 0 with boundary
conditions, then either y is a solution of the differential
for all
y (t ) − y0 (t ) ≤ k (ε ) And lim k (ε ) = 0
ε →0
equation
Those previous results were extended to the Hyers-Ulam
stability of linear differential equations of first order and
higher order with constant coefficients in [8, 18] and in
[19] respectively.
Rus investigated the Hyers-Ulam stability of differential
and integral equations using the Granwall lemma and the
technique of weakly Picard operators [15, 16].
Miura et al [13] proved the Hyers-Ulam stability of the
first-order linear differential equations
y '(t ) + g (t ) y (t ) = 0 ,
Where g (t ) is a continuous function, while Jung (4)
proved the Hyers-Ulam stability of differential equations
of the form
ϕ (t ) y '(t ) = y (t )
ϕ ( f , y, y ',..., y ( n ) ) = 0 …… ……(2)
or y ( x ) ≤ kε for any x ∈ [ a, b ] ,
y' = λy
Li and Shen [6] proved the stability of non-homogeneous
linear differential equation of second order in the sense of
the Hyers and Ulam
constant. Then, we say that (2) has super stability with
boundary
conditions.
Preliminaries
Lemma 1[10]. Let
max
While Gavaruta et al [1] proved the Hyers- Ulam stability
of the equation
y ''+ β ( x ) y ( x ) = 0
The recently, introduced notion of super stability
[10,11,12,13] is utilized in numerous applications of the
automatic control theory such as robust analysis, design of
static output feedback, simultaneous stabilization, robust
stabilization, and disturbance attenuation.
Recently, Jinghao Huang, Qusuay H. Alqifiary, Yongjin Li[3]
established the super stability of differential equations of
second order with boundary conditions or with initial
conditions as well as the super stability of differential
equations
of
higher
order
in
the
form
with
( n −1)
initial
conditions,
(a) = 0
This study aimed to extend the super stability of second
order to the third order linear ordinary differential
homogeneous equations with boundary condition.
|
2
max y ''( x) .
8
Proof
Let
M
max { y ( x) : x ∈ [ a, b ]} Since
=
y (a ) = 0 = y (b ) , there exists x0 ∈ (a, b) such that
y ''(δ )
( x0 − a ) 2 ,
2!
y ''(η )
y (b) = y ( x0 ) + y '( x0 )(b − x0 ) +
(b − x0 ) 2 ,
2!
Where δ , η ∈ ( a, b )
y ( a ) = y ( x0 ) + y '( x0 )( x0 − a ) +
Thus
y ''(δ ) =
y ''(η ) =
2M
( x0 − a )
( b − x0 )
Unique Paper ID – IJTSRD29149
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( x0 − a )
2
,
2

2M
2
2M
In the case x0 ∈  a,
with boundary and initial conditions.
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y ∈ c 2 [a, b] , y (a ) = 0 = y (b ) ,then
(b − a )
y ( x) ≤
y ''+ p ( x ) y '+ q ( x ) y + r ( x ) = 0
y (a) = y '(a) = ... = y
k is a
y ( x0 ) = M . By Taylor’s formula, we have
Motivation of this study comes from the work of Li [5]
where he established the stability of linear differential
equations of second order in the sense of the Hyers and
Ulam.
y ( n ) ( x) + β ( x ) y ( x ) = 0
where
≥

2M
( a + b )  , we have
(b − a )
2
2
=


8M
(b − a )
2
4
a +b 
In the case x0 ∈ 
, b  , we have
 2

2M
2M
8M
≥
=
2
2
2
( b − x0 ) ( b − a ) ( b − a )
4
8M
8
=
So, max y ′′( x ) ≥
max y ( x)
2
2
(b − a ) (b − a )
Therefore, max
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y ( x)
(b − a )
≤
8
2
max y ''( x)
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In (2011) the three researchers Pasc Gavaruta, Soon-Mo,
Jung and Yongjin Li, investigate the Hyers-Ulam stability of
second order linear differential equation
y ''( x ) + β ( x) y ( x ) = 0 ………….. (3)
with boundary conditions y ( a ) = y (b) = 0
where,
y ∈ c 2 [ a, b] , β ( x) ∈ c [ a, b ] , −∞ < a < b < +∞
Definition: We say (3) has the Hyers-Ulam stability with
boundary conditions
y (a ) = y (b) = 0 if there exists a positive constant K
with the following property:
For every
ε > 0, y ∈ c 2 [ a, b] ,
if
y ''( x) + β ( x) y ( x) ≤ ε ,
then
there
exists
some
z ∈ c [ a, b ] satisfying
z ''( x ) + β ( x ) z ( x ) = 0
And z ( a ) = 0 = z (b ) , such that
y ( x ) − z ( x) < K ε
≤
Theorem 1[1]. Consider the differential equation
y ''( x ) + β ( x) y ( x ) = 0 --------------------- (4)


8
then the equation (4) above has the super stability with
boundary condition
y ( a ) = y (b ) = 0
For
ε > 0, y ∈ C 2 [ a, b ]
every
if
,
there exists
Such that
{ y( x) } : x ∈ [ a, b] , since y (a) = 0 = y (b) ,
x0 ∈ (a, b)
y ( x0 ) = M . By Taylor formula, we have
y ''(δ )
( x0 − a ) 2 ,
2!
y ''(η )
y (b) = y ( x0 ) + y '( x0 )(b − x0 ) +
(b − x0 ) 2 ,
2!
Where δ ,η ∈ ( a, b )
y ( a ) = y ( x0 ) + y '( x0 )( x0 − a ) +
Thus y ''(δ ) =
2M
( x0 − a )
2
,
2M
y '' (η ) =
( x0 − b) 2


On the case x0 ∈  a,
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z0 ( x ) = 0
is
a
solution
of
y ''( x ) − β ( x ) y ( x ) = 0 with the Boundary conditions
y (a) = 0 = y(b) .
y ( x ) − z0 ( x ) ≤ K ε
Hence the differential equation y ''( x ) + β ( x ) y ( x ) = 0
has the super stability with boundary condition
y (a) = 0 = y(b)
y ''( x) + β ( x) y ( x) ≤ ε and y (a ) = 0 = y (b )
Let M = max


8
Obviously,
2
Proof:
(b − a)2
(b − a) 2
ε+
max β ( x) max y ( x)
8
8
2
y ∈ c 2 [ a, b] , β ( x) ∈ c [ a, b ] , −∞ < a < b < +∞
(b − a )
(b − a)2
 max y ''( x) − β ( x) y ( x) + max β ( x) ma
8 
(b − a )
Let k =
 ( b − a )2

8 1 −
max β ( x) 
With boundary conditions y ( a ) = y (b) = 0
max β ( x) <
(b − a ) 2
max y ''( x)
8
Thus
max y ( x) ≤
2
If
max y ( x) ≤
Therefore
And y ( a ) = y (b) = 0 ,
Where
2M
2M
8M
≥
=
2
2
(b − a )
( x0 − a )
(b − a ) 2
4
 ( a + b) 
On the case x0 ∈ 
, b  we have
 2

2M
2M
8M
≥
=
(b − x0 ) 2 (b − a ) 2 (b − a )2
4
8M
8
So max y ''( x) ≥
=
max y ( x)
2
(b − a)
(b − a )2
In (2014) the researchers J. Huang, Q. H. Aliqifiary, Y. Li
established the super stability of the linear differential
equations.
y ''( x) + p( x) y '( x) + q( x) y( x) = 0
With boundary conditions y (a) = 0 = y (b)
Where
y ∈ c 2 [ a , b ] , p ∈ c1 [ a , b ] , q ∈ c 0 [ a , b ] , −∞ < a < b < +∞
Then the aim of this paper is to investigate the super
stability of third-order linear differential homogeneous
equations by extending the work of J. Huang, Q. H.
Aliqifiary and Y. Li using the standard procedures of them.
Lemma( 2) .Let
max
( a + b) 
, we have
2 
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y ∈ c3 [ a, b] and y (a ) = 0 = y (b ) , then
(b − a )
y ( x) ≤
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3
max y '''( x)
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Proof
{
[
Let M = max y ( x) : x ∈ a, b
]} since
y (a ) = 0 = y (b ) there exists:
x0 ∈ ( a, b ) Such that y ( x0 ) = M
By Taylor’s formula we have
y ( a ) = y ( x0 ) + y '( x0 )( x0 − a ) +
y ''( x0 )
y '''(δ )
2
3
( x0 − a ) +
( x0 − a ) ,
2!
3!
y (b) = y ( x0 ) + y '( x0 )(b − x0 ) +
y ''( x0 )
y '''(η )
(b − x0 ) 2 +
(b − x0 )3 ,
2!
3!
⇒ y (a ) ≤ y ( x0 ) + y '( x0 ) ( x0 − a ) +
y ''( x0 )
y '''(δ )
2
3
( x0 − a ) +
( x0 − a ) ,
2!
3!
And ⇒ y (b) ≤ y ( x0 ) + y '( x0 ) (b − x0 ) +
Where
y ''( x0 )
y '''(η )
(b − x0 )2 +
(b − x0 )3 ,
2!
3!
δ ,η ∈ ( a, b )
Thus y '''(δ ) =
And y '''(η ) =
3! M
6M
=
3
( x0 − a )
( x0 − a )3
3! M
6M
=
3
(b − x0 )
(b − x0 )3


For the case x0 ∈  a,
a + b
a+b
that is a < x0 ≤
,we have

2
2 
6M
6M
6M
48M
≥
=
=
3
3
3
(b − a )
( x0 − a )
(b − a )3
 a+b

−
a


8
 2

a+b
a + b 
, b  ,that is
≤ x0 < b we have
2
 2

And for the case x0 ∈ 
6M
6M
6M
48M
≥
=
=
3
3
3
3
(b − x0 )
(b − a ) (b − a )
b−a


8
 2 
⇒
6M
( b − x0 )
3
≥
48M
(b − a )
3
Thus
max y '''( x) ≥
48M
48
=
max y ( x)
3
(b − a )
(b − a)3
from
max y '''( x) ≥
48
max y ( x)
(b − a )3
⇒ max y ( x) ≤
(b − a)3
max y '''( x)
48
Theorem 2. Consider y '''( x ) + m ( x ) y ''( x ) + p ( x ) y '( x ) + q ( x ) y ( x ) = 0 ------------------ (5)
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with boundary conditions
where
y ( a ) =0= y ( b )
y ∈ c3 [ a, b ] , m ∈ c 2 [ a, b ] , p ∈ c ' [ a, b] , q ∈ c 0 [ a, b ] − ∞ < a < b < +∞
+ − ′ + ′ − 2′ + − 2 − <
If -------(6)
y (a ) = 0 = y ( b )
Then (5) has the super stability with boundary conditions
Proof
Suppose that
y ∈ c 3 [ a, b ] satisfies the inequality:
y '''( x) + m( x) y ''( x) + p( x) y '( x) + q( x) y ( x) ≤ ε ------------ (7) for some ε > 0
Let
U ( x ) = y '''( x ) + m( x ) y ''( x ) + p ( x) y '( x) q ( x ) y ( x) --------- (8)
For all
x ∈ [ a, b] and define z ( x ) by
x
y ( x ) = z ( x )e
−
1
p (τ ) dτ
2
∫
a
--------- (9)
And by taking the first, second and third derivative of (9)
That is
x
1
 ′ 1  −2
′
y ( x) =  z − zp  e ∫ p (τ )dτ
2 

a
1
x
− p (τ ) dτ
1
1

 2∫
y′′( x) =  z ′′ − z ′p − zp′ + zp 2  e a
2
4


x
1
1
1
1
1
1
1
1


y '''( x) =  z '''− z '' p − z '' p − z ' p′ + z ′p 2 − z ′p′ − zp′′ + zpp′ + z ′p 2 + zp − zp 3  e
2
2
2
2
4
4
2
8


−
1
p (τ ) dτ
2
∫
a
And by substituting (9) and its first, second and third derivatives in (8) we get
1
1
1
1
1
1
1
1

u ( x) =  z '''− z '' p − z '' p − z ' p '+ z ' p 2 − z ' p '− zp ''+ zpp '+ z ' p 2 + pz − zp 3 +
2
2
2
2
4
4
2
8

x
1
1


m  z ''− z ' p − zp '+ zp 2  +
2
4




=  z '''−
1 

p  z '− zp  + zq
2 

]e
−
1
p (τ ) dτ
2
∫
a
1
1
1
1
1
1
1
1
z '' p − z '' p − z ' p '+ z ' p 2 − z ' p '− zp ''+ zpp '+ z ' p 2 + zp − zp 3 +
2
2
2
2
4
4
2
8
x
mz ''− z ' mp −


=  z '''+ mz ''−
1
1
1
zmp '+ zmp 2 + pz '− zp 2 + qz
2
4
2
]e
−
1
p (τ ) dτ
2
∫
a
3
3
3
1
1
1
1
z '' p + z ' p − z ' mp − z ' p '+ z ' p 2 + qz + zp + zpp '− zmp '− zp ''+
2
2
4
2
4
2
2
x
1
1
1
zmp 2 − zp 2 − zp 3
4
2
8
=
[
3

z '''+  m −
2

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]e
−
1
p (τ ) dτ
2
∫
a
3
3 


p  z ''+  p − mp − p '+ p 2  z '+
2
4 


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x
1
1
1 3

2
2
 q + ( p − p '') + ( pp '− 2mp '+ mp − 2 p ) − p  z
2
4
8 

]e
−
1
p (τ ) dτ
2
∫
a
3 2 3
p − p '+ p
3
4
2
Now choose m = p and m =
2
p
By doing this the coefficients of z ''( x ) and z '( x ) will vanish.
To check whether the relation holds or not, for
3 2 3
p − p '+ p
3
4
2
p=
2
p
⇒
3 2 3 2 3
p = p − p '+ p
2
4
2
⇒
3
3
p ' = p − p2
2
4
⇒
3
p' =
2
 3
p 1 −
 4
3 dp
 3
= p 1 −
2 dx
 4
⇒

p


p

dp
2
= dx using partial fraction
 3  3
p 1 − p 
 4 
3
1
1
+ 4 =
Since
3
3
p
1 − p p 1 −
4
 4

p

3 

1
4  dp = 2 dx
 +
--------(*)
3 
3
 p 1− p 

4 
By integrating both sides of (*) we have
ln p − ln 1 −
⇒ ln
3
2
p = x + c1
4
3
p
2
= x + c1
3
1− p 3
4
2
x
p
⇒
= Ce 3
3
1− p
4
2
x
3
 3 
p = Ce 1 − p 
 4 
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= Ce
2
x
3
2
x
3
− Cpe 3
4
2
2
x
x
 3 2x 
3
p + Cpe 3 = Ce 3 = p  1 + Ce 3 
4
 4

2
p=
Ce 3
x
3
1 + Ce
4
⇒ m=
2
x
3

2
x
3  Ce 3
m= 
2
2
3 3x
 1 + Ce

4
3
p
2






1
x
1
1
1 3  − 2 ∫a p (τ ) dτ

2
2
z
'''
+
q
+
(
p
−
p
'')
+
(
pp
'
−
2
mp
'
+
mp
−
2
p
)
−
p  ze
= u ( x)
Then

2
4
8 

Then from the inequality (7) we get
x
1
p (τ ) dτ
2
∫
u ( x) e
a
1
1
1 

z '''+  q + ( p − p '') + ( pp '− 2mp '+ mp 2 − 2 p 2 ) − p3  z =
2
4
8 

x
≤ εe
1
p (τ ) dτ
2
∫
a
x
From the boundary condition y ( a ) = 0 = y (b) and
y ( x ) = z ( x )e
−
1
p (τ ) dτ
2
∫
a
We have z ( a ) = 0 = z (b )
Define
1
1
1
β ( x) = q + ( p − p '') + ( pp '− 2mp '+ mp 2 − 2 p 2 ) − p3
2
4
8
x
Then
z '''( x) + β z ( x) = u ( x) e
1
p (τ ) dτ
2
∫
a
x
≤ εe
1
p (τ ) dτ
2
∫
a
Using lemma (2)
max z ( x) ≤
≤
≤
(b − a)
48
3
48
max z '''( x)
3
 max z '''( x) + β z ( x) + max β max z ( x) 
48
(b − a )
(b − a )
3
 1 ∫ p (τ ) dτ 
( b − a ) max β max z ( x)
 2

max e a
ε
+

48




x
3
x
1
p (τ ) dτ
2
∫
Since
max e
a
< ∞ on the interval [ a, b]
Hence, there exists a constant K > 0 such that
z ( x) ≤ k ε For all x ∈ [ a, b]
x
−
More over
@ IJTSRD
max e
|
1
p (τ ) dτ
2
∫
a
< ∞ on the interval [ a, b] which implies that there exists a constant such that K ' > 0
Unique Paper ID – IJTSRD29149
|
Volume – 3 | Issue – 6
|
September - October 2019
Page 703
International Journal of Trend in Scientific Research and Development (IJTSRD) @ www.ijtsrd.com eISSN: 2456-6470
 1x

y ( x) = z ( x) exp  − ∫ p(τ )dτ 
 2a


 1x
 
≤ max exp  − ∫ p(τ )dτ   k ε ≤ k ' ε
 2a
 

⇒ y( x) ≤ k ' ε
Then y '''( x ) + m ( x ) y ''( x ) + p ( x ) y '( x ) + q ( x ) y ( x ) = 0 has super stability with boundary conditions
y (a ) = 0 = y (b )
Example
Consider the differential equation below

2
x

3
e3
y '''+ 
2
2
3 3x
e
1
+


4


2

 e3x
 y ''+ 
2
3 3x


e
1
+




4


 y '+ y = 0 ----------- (10)



With boundary conditions y ( a ) = 0 = y (b )


2
2
x
x


3
3
e
e3
3
2
 ∈ c [ a, b] ,
∈ c1 [ a, b] ,1∈ c0 [ a, b] , −∞ < a < b < +∞
Where y ∈ c [ a, b] , 
2
2
2  3 3x 
3 x
1 + e3
 1+ e 
4
 4 


2
2

x
3
 e3 x
1 e
−
If max 1 + 
2
2
2
3 3x  3 3x
 1+ e
e
1
+

 4
 4


2
x

3
e3

2  3 23 x
 1+ e
 4

2
 e 3 x

2
 1 + 3 e 3 x

 4
2

′′ 

2
  1  e3x
 + 
2
  4  1+ 3 e3x


 
 4



2

 e3x
 − 2
2

 1+ 3 e3 x



 4
2

2
  e3 x

2
  1+ 3 e3x

 4


2
 1  e3 x
 − 
2
 8  1+ 3 e3x



 4
′ 
2

 e3x
 − 3
2

 1+ 3 e3 x



 4

2
 e 3 x

2
 1 + 3 e 3 x

 4
′

 +



3


48
 <
3 --------- (11)
b − a)

(


Then (10) has super stability with boundary conditions y ( a ) = 0 = y ( b )
Suppose that
y ∈ c 3 [ a, b ] satisfies the inequality

2
x
3  e3
y′′′ + 
2  3 23 x
 1+ e
 4


2

 e3 x
 y′′ + 
2

 1+ 3 e3x



 4

2
x

3
e3
Let v(x) = y ′′′ + 
2
2
3 3x
 1+ e

4
For all


 y′ + y ≤ ε ,for some ε > 0 --------- (12)





2

 e3x
 y′′ + 
2

 1+ 3 e3x




4


 y′ + y ------- (13)



x ∈ [ a, b] and define z ( x ) by
@ IJTSRD
|
Unique Paper ID – IJTSRD29149
|
Volume – 3 | Issue – 6
|
September - October 2019
Page 704
International Journal of Trend in Scientific Research and Development (IJTSRD) @ www.ijtsrd.com eISSN: 2456-6470
−
y ( x ) = z ( x )e
1
2
2
x
∫
a
e3
x
2
dτ
3 x
1+ e 3
4
----- (14)


2
x


1
e3
y′ =  z ′ − z 
2
2 
3 3x

1
+
e



4



2

x

3
e

y′′ =  z′′ − z′ 
2
3 3x


 1+ e

 4

2
x
  − 1 ∫ e 3 2 dτ
  2 a 1+ 3 e 3 x
4
e




x


2
 1  e3x
− z
2
 2  1+ 3 e3x



 4
′

2
 1  e3 x
 + 
2
 4  1+ 3 e3x



 4




2
2
x


 e3 x
1  e3
y ′′′ =  z ′′′ − z ′′ 
 − z ′′ 
2
2
2 
3 3x 

 1+ 3 e3 x
1
+
e





4


4


2
x
1  e3
z′ 
2  3 23 x
 1+ e
 4
′

2
 1  e3x
 − z
2
 2  1+ 3 e3 x



 4

2
x

1
e3
z′ 
4  3 23 x
 1+ e
 4


2
 1  e3 x
 + 
2
 2  1+ 3 e3 x



 4
2


2

 e3x
 − z′ 
2

1+ 3 e3x




4






2
 1 x e3x
 − 2 ∫a 3 2 x
 e 1+ 4 e 3




2
'


2
 1  e3x
 + z′ 
2
 2  1+ 3 e3x




4
′′

2
 1  e3 x
 + z
2
 4  1+ 3 e3x



 4
′ 
2
  e3 x

2
  1+ 3 e3 x

 4


2
x


1
e3
z − z
8  3 23 x


1+ e

 4






3
−
)e
1
2
2
x
∫
a
e3
2


 −





+



x
2
dτ
3 x
1+ e 3
4
By substituting (14) and its first, second and third derivatives in (13) we get

 



2
2

 3  e3 x  3  e3 x  
 z′′′ +  
− 
  z′′ +
2
2

 2  3 3x  2  3 3x 
 1+ e  
  1+ e 


 4

  4



2
2

x
x
3

3
e
3
e

 3 2x − 2  3 2x

 1+ e3
 1+ e3
 4
 4




2
2
  e3x  3  e3x

− 
2
2
  1+ 3 e3 x  2  1+ 3 e3x



 4

 4


 
2
2

x
x

3


1
e
e3

+ 1 + 
−
2  3 23 x   3 23 x

 1+ e   1+ e

 4
  4


2
x
3  e3

+
2
2
3 x
1 + e3

4
@ IJTSRD
|

2
 e3x

2
3 x

1 + e3

4
′

2
 3  e3x
 + 
2
 4  1+ 3 e3x



 4






2


 z′




′′


′ 
2
2
2
x
x
 1  e3
  e3

 e3x
 + 

 − 3
2
2
2
 4  1+ 3 e3 x   1+ 3 e3 x 
 1+ 3 e3x






 4
 4

 4
2


2

 e3x
 − 2
2
3 x



1 + e3


4
Unique Paper ID – IJTSRD29149
2


2
x

1  e3
 − 
2
8
3 x


1 + e3


4
|







2
 e 3 x

2
 1 + 3 e 3 x

 4
2
3
−
) z  e
Volume – 3 | Issue – 6
1
2
x
∫
a
|
e3
′





x
2
dτ
3 x
1+ e 3
4
September - October 2019
Page 705
International Journal of Trend in Scientific Research and Development (IJTSRD) @ www.ijtsrd.com eISSN: 2456-6470
Since z ' and z ′′




2
2



x
x
3

e3

 1 e
v( x) =  z ′′′ + 1 + 
−
2
2
x
2
3
 1+ 3 e3 x
3


 1+ e

 4

 4





2
x
3  e3
+ 
2
2
3 3x
1+ e

4

2
  e3 x

2
3 3x

1+ e

4

′′ 

2
  1  e3x
 + 
2
  4  1+ 3 e3x


 
 4

2


2

 e3x
 − 2
2
3 3x



1+ e


4

2
  e3 x

2
  1+ 3 e3 x

 4
2


2
 1  e3x
 − 
2
3 3x
 8

1+ e


4






′ 
2

 e3x
 − 3
2

 1+ 3 e3x



 4
2
3
) z  e
1
−
2
x
∫
a
e3

2
  e3 x

2
  1+ 3 e3x

 4
′





x
2
dτ
3 x
1+ e 3
4
Then from inequality (12) we get



2
2


x
x
3

e3
 1 e
z ′′′ + 1 + 
−
2
2
2
3 3x 
3 x

 1+ e
 1+ e3



4
4



2
x
3  e3
+ 
2
2
3 3x
1+ e
4


2
  e3x

2
3 3x

1+ e
4


′′ 

2
  1  e3 x
 + 
2
  4 1+ 3 e3x


 

4

2


2

 e3x
 − 2
2
3 3x



1+ e
4



2
 e 3 x

2
 1 + 3 e 3 x


4
2


2
 1  e3x
 − 
2
3 3x
 8

1+ e
4








′ 
2

 e3 x
 − 3
2

1+ 3 e3x




4

2
 e 3 x

2
 1 + 3 e 3 x


4
′





3
)z




2
2
 1 x e3x

 1 x e3x



= v( x) exp  ∫
d
τ
≤
exp
d
τ
2
∫ 3 2 x  ε
3 3x
2 a

2a
 1+ e

 1 + e3


4


4

From the boundary condition y ( a ) = 0 = y (b) and


2
 1 x e3x

 we have z ( a ) = 0 = z (b )
y ( x) = z ( x) exp  − ∫
d
τ
2
3 3x
 2a

1+ e



4

Define:


2
2

x
3
 e3x
1 e
−
β = 1+ 
2
2
2
3 3x  3 3x
 1+ e
1
+
e

 4
 4


2
x
3  e3
+ 
2
2
3 3x
1+ e

4
@ IJTSRD
|

2
  e3x

2
3 3x

1+ e

4

′′ 

2
  1  e3 x
 + 
2
  4  1+ 3 e3x


 
 4

2


2

 e3x
 − 2
2
3 3x



1+ e


4
Unique Paper ID – IJTSRD29149
2

2
 e 3 x

2
 1 + 3 e 3 x

 4


2
 1  e3x
 − 
2
3 3x
 8

1+ e


4
|






′ 
2

 e3x
 − 3
2

 1+ 3 e3x



 4

2
 e 3 x

2
 1 + 3 e 3 x

 4
′





3
Volume – 3 | Issue – 6
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September - October 2019
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



2
2
 1 x e3x

 1 x e3x

Then z ′′′ + β z ( x ) = v ( x ) exp  ∫
dτ  ≤ exp  ∫
dτ  ε
2
2
 2 a 1+ 3 e3 x

 2 a 1+ 3 e3x









4
4
using lemma (2)
max z ( x) ≤
(b − a )
≤
(b − a )
48
48
3
max z '''( x)
3
 max z′′′ + β z ( x) + max β max z ( x) 
48
≤
(b − a )
3



2
3

 1 x e3x
 
b − a)
(
max exp  ∫
dτ   ε +
max β max z ( x)
2
x
2
48
3


a


 1 + e3


4

 



2

 1 x e3x
 
  < ∞ on the interval [ a, b]
since max exp  ∫
τ
d
2
x
2
3


a

 1 + e3



4
 
Hence there exists a constants
k > 0 such that z ( x) ≤ kε


2
 1 x e3x


y ( x) = z ( x) exp  − ∫
d
τ
2
3 3x
 2a

1+ e



4




2

 1 x e3x
 

≤ max exp − ∫
dτ   k ε ≤ k ' ε
2
3 x
 2a


1 + e3




4
 
⇒ y ( x) ≤ k 'ε

2
x

3
e3
Then the differential equation y′′′ + 
2
2
3 3x
1+ e

4


2

 e3x
 y′′ + 
2

1+ 3 e3x




4


 y′ + y = 0



Has the super stability with boundary condition y ( a ) = 0 = y (b ) on closed bounded interval



2
2


x
3
 e3 x
 1 e
max  1 + 
−
2
2
 2  1 + 3 e 3 x  1 + 3 e 3 x

 4
 4
To show the



′′ 

2
  1  e3 x
 + 
2
  4  1+ 3 e3x


 
 4

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
2
  e3 x

2
  1+ 3 e3x

 4
Volume – 3 | Issue – 6
|
[ a, b ]
′ 
2

 e3 x
 − 3
2

 1+ 3 e3x



 4

2
  e3 x

2
  1+ 3 e3x

 4
September - October 2019
′

 +



Page 707
International Journal of Trend in Scientific Research and Development (IJTSRD) @ www.ijtsrd.com eISSN: 2456-6470

2
x
3  e3

2
2
3 3x
1+ e

4

2
 e3x

2
3 3x

1+ e

4
2


2

 e3x
 − 2
2
3 3x



1+ e


4
2


2
 1  e3x
 − 
2
3 3x
 8

1+ e


4






3
}<
48
(b − a )
3
− − − − − −(**)
To make our work simple we simplify the expression (**)
Then the simplified form of (**) is
2
3
4
2
2
2
2

x
x 
x 
x 



5
15
35
33

e3 +  e3  +  e3  −
 e3 
6
8
32
128






max  1 +
4
2
 3 3x 

1 + e 

2






48
<
3 ---------- (***)
 (b − a)


Figure:1
2
3
4
2
2
2
2

5 3 x 15  3 x  35  3 x 
33  3 x 

e + e  + e  −
e 
6
8
32 
128 




The graph which shows the max  1 +
4
2
 3 3x 

1+ e 


 2


Conclusion
In this study, the super stability of third order linear
ordinary differential homogeneous equation in the form of
y ′′′( x ) + m ( x ) y′′( x ) + p ( x ) y ′( x ) + q ( x ) y ( x ) = 0 with
boundary condition was established. And the standard
work of JinghaoHuang, QusuayH.Alqifiary, Yongjin Li on
investigating the super stability of second order linear
ordinary differential homogeneous equation with
boundary condition is extended to the super stability of
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Unique Paper ID – IJTSRD29149
|



48
<
3
 (b − a )


third order linear ordinary differential homogeneous
equation with Dirchilet boundary condition.
In this thesis the super stability of third order ordinary
differential homogeneous equation was established using
the same procedure coming researcher can extend for
super stability of higher order differential homogeneous
equation.
Volume – 3 | Issue – 6
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September - October 2019
Page 708
International Journal of Trend in Scientific Research and Development (IJTSRD) @ www.ijtsrd.com eISSN: 2456-6470
Reference
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