Differential Equations Third Order Inhomogeneous Linear with Boundary Conditions

International Journal of Trend in Scientific Research and Development (IJTSRD)
Volume 8 Issue 1, January-February 2024 Available Online: www.ijtsrd.com e-ISSN: 2456 – 6470
@ IJTSRD | Unique Paper ID – IJTSRD63458 | Volume – 8 | Issue – 1 | Jan-Feb 2024 Page 630
Differential Equations Third Order
Inhomogeneous Linear with Boundary Conditions
Ghulam Hazrat Aimal Rasa
Department of Mathematics, Kabul Education University, Kabul, Afghanistan
ABSTRACT
Considering the importance of teaching linear differential equations,
it can be said that every physical and technical phenomenon, which is
expressed and modeled in mathematical sciences, is a differential
equation. Differential equations are essential part of contemporary
comparative mathematics that covers all fields of physics (heat,
mechanics, atoms, electronics, magnetism, light and waves), many
economic subjects, engineering subjects, natural problems,
population growth and technical problems today. In this article, we
will consider the theory of linear inhomogeneous differential
equations of the third order with boundary conditions and the
transformation of coefficients into multiple ( )
p x functions. In the
field of differential equations, a boundary value problem with a set of
additional constraints is called boundary value problem. The solution
of this boundary value problem is actually a solution for the
differential equation with the given constraints, which actually
satisfies the conditions of the boundary value problem. Differential
equation problems with boundary conditions are similar to initial
value problems. A boundary value problem with conditions defined
on the boundaries is an independent variable in the equation, while an
initial value problem is defined as the same condition that has the
value of the independent variable and this value is less than the limit,
hence the term value is initial and the initial value is the amount of
data that matches the minimum or maximum input, internal, or output
value specified for a system or component. When the boundaries of
the boundary values in the solution of obtaining the constants of the
third order differential equation 1
D , 2
D and 3
D are determined, the
failure to obtain the constants is called the boundary problem. We
solve this problem by considering the given conditions for the real
Green's function. Every real function is a solution of a set of linear
differential equations, and the values of its boundary value depend on
the intervals.
How to cite this paper: Ghulam Hazrat
Aimal Rasa "Differential Equations
Third Order Inhomogeneous Linear with
Boundary Conditions" Published in
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and Development
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KEYWORDS: Green's function,
boundary value problem, private
solution, general solution
1. INTRODUCTION
Differential equations are one of the most interesting
and widely used mathematical topics that have
attracted the attention of many researchers.
Differential equations are used in various fields
including physics; It is especially useful in the
movement of weights attached to it, springs, electric
circuits, and free vibrations. In mathematics, in the
field of differential equations, the boundary value
problem of a differential equation with an additional
set of constraints is called the boundary condition
problem, and the solution that satisfies the given
conditions also satisfies [1,2,3]. boundary value
problems arise in several branches of physics because
each equation has a differential body. Wave equation
problems, such as determining normal modes, are
often referred to as boundary value problems.
Another big group of important boundary value
problems are Sturm-Liouville problems. The analysis
of these problems includes special functions and
Green's functions of a differential equation [3, 4].
In this article, discussions are discussed to express
and understand more about the problem of third-order
inhomogeneous linear differential equations with
boundary conditions and obtaining Green's function.
IJTSRD63458

International Journal of Trend in Scientific Research and Development @ www.ijtsrd.com eISSN: 2456-6470
In the space 2 (0,1)
L of an inhomogeneous linear
differential equation of the third order with boundary
conditions, we consider the boundary problem and on
that initial function is obtained by a linear differential
equation of the third order with constant coefficients
[2].
Suppose we consider the third order differential
equation as follows:
(3) (1)
1 0
( ) ( ) ( ) ( ) ( ) ( )
y x P x y x P x y x f x
+ + = (1)
Where ( )
0
P x and ( )
1
P x many functions on intervals
[ ]
0,1 , the number 3 expresses the order of the
differential equation.
In this section, we consider the known features of
these functions with the following boundary
conditions:
( ) ( ) ( )
(0) (1) 0, 1,2,3
j j
j j j
U y y y j
γ γ
α β
= + = = (2)
It should be known that
1 2 3
0, 1, 2
γ γ γ
= = =
is remembered.
From the statement of the above problem, the
following questions can be reached:
Research questions
1. How can we solve the inhomogeneous linear
differential equation of the third order
(3) (1)
1
( ) ( ) ( ) ( )
y x P x y x f x
+ = with boundary
conditions ( ) ( ) ( )
(0) (1) 0, 1,2,3
j j
j
U y y y j
γ γ
= + = = ?
2. How can we solve the private and general solution
of the inhomogeneous third-order linear differential
equation
(3) (2) (1)
2 1 0
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
y x P x y x P x y x P x y x f x
+ + + = if
it is with 2 ( ) 0,
P x ≠ 1 0
( ) 0, ( ) 0
P x P x
≠ ≠ and given
boundary conditions
( ) ( ) ( )
(0) (1) 0, 1,2,3
j j
j
U y y y j
γ γ
= + = = ?
3. How can we solve the inhomogeneous linear
differential equation of the third order with the
conditions of the boundary problem
( ) ( ) ( )
(0) (1) 0, 1,2,3
j j
j
U y y y j
γ γ
= + = = in the
determined area?
The opinion framework is based on third order
differential equations with boundary conditions,
Green's function, eigenfunctions and eigenvalues to
obtain general solution and private solution on
boundary conditions, Wronski determinant and
differential operators.
This research is divided into six basic parts:
introduction, review of scientific works, basic
concepts, research findings, controversy and
conclusion.
2. Literature review
Differential equations have been developed for nearly
300 years, and the relationship between the
transformations of functions and the derivatives of
functions, so its history naturally goes back to the
discovery of the derivative by the English scientist
Isaac Newton between the years (1642-1772). And
the German Gottfried Leibniz worked on differential
equations, including first-order differential equations,
in the years (1646-1716). Jacob proposed Bernoulli's
differential equation in 1674, but he was unable to
prove it until Euler proved it in 1705. Sturm-Liouville
theorized the boundary problem with the first
boundary in linear differential equations. And its
applications, the classical Sturm-Liouville theory,
named after Jacques Francois Sturm and Joseph
Liouville, was proposed between (1855-1803) and in
(1809-1882), the theory of linear differential
equations was formed in the second order. In 1969,
the Russian scientist Naimark wrote in his book
Linear Differential Functions about the Green's
function for solving differential equations with
boundary conditions. According to the theorems of
Mikhailov and Kesselman, the boundary conditions
are often strictly regular and defined [4]. Therefore,
the eigenvalues of the asymptotic operator are simple
and distinct, there is a positive number such as δ,
which are separated from each other by a greater
distance δ for both eigenvalues of the function [3]. It
is also concluded from the works [1, 2, 3, 7, 11, 12,
13, 14, 15] that the system of eigenfunctions and
related functions form a basis Res in the space.
In recent years, many pure mathematical scientists
have worked in the field of obtaining Green's function
for linear differential equations, including the Kazakh
scientist Kanguzhin in 2019, who published an article
entitled "Getting Green's function for second-order
linear differential equations" [4, 6].
3. Elementary Basic
The general form of inhomogeneous linear
differential equations can be written as follows,
considering differential operators:
( ) ( ) ( )
L y y x f x
λ
= + (3)
Considering the system of high-order linear
differential equations of the general solution of
equation (1), (2), we can consider an initial function
as follows:
0 1 1 2 2 3 3
( ) ( ) ( ) ( ) ( )
y x y x D x D x D x
ϕ ϕ ϕ
= + + + (4)

Where
0
0
( ) ( , ) ( )
x
y x g x t f t dt
= 
0( )
у x is the homogeneous solution of the above equation and 1 2 3
( ), ( ), ( )
x x x
ϕ ϕ ϕ The main system of solving
the equation with homogeneous conditions 1 2 3
( ) 0, ( ) 0, ( ) 0
L L L
ϕ ϕ ϕ
= = = is one of the inhomogeneous
boundary conditions
( 1)
(0)
k
j k j
ϕ δ
−
= function ( , )
g x t It is determined by the following formula, which can be
called Green's function [9].
( , )
( , )
( )
P x t
g x t
W t
=
Where



≠
=
=
j
k
j
k
kj
,
0
,
1
δ and )
(t
W determinant Wronski
1 2 3
(1) (1) (1)
1 2 3 1 2 3
(2) (2) (2)
1 2 3
( ) ( ) ( )
( , , ) ( ) ( ) ( )
( ) ( ) ( )
y t y t y t
W t t t y t y t y t
y t y t y t
=
And it should be known that ( , )
P x t is equal to:
1 2 3
(1) (1) (1)
1 2 3
1 2 3
( ) ( ) ( )
( , ) ( ) ( ) ( ) .
( ) ( ) ( )
y t y t y t
P x t y t y t y t
y x y x y x
=
So you should know that ( , ) ( , )
g x t P x t
= so ( , )
g x t can be defined from the following formula.
1 2 3
(1) (1) (1)
1 2 3
1 2 3
( ) ( ) ( )
( , ) ( ) ( ) ( ) .
( ) ( ) ( )
y t y t y t
g x t y t y t y t
y x y x y x
=
From here we can propose a specific inhomogeneous solution as follows
1 2 3
(1) (1) (1)
0 1 2 3
0
1 2 3
( ) ( ) ( )
( ) ( ) ( ) ( ) ( )
( ) ( ) ( )
x
y t y t y t
y x y t y t y t f t dt
y x y x y x
= 
The inhomogeneous solution function 0 ( )
y x is equation (1), (2) and for its correctness, I search the first, second
and third order derivatives and establish the proposed third order equation (1).
1 2 3 1 2 3
(1) (1) (1) (1) (1) (1) (1)
0 1 2 3 1 2 3
(1) (1) (1)
0
1 2 3 1 2 3
( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) ( )
x
y t y t y t y x y x y x
y x y t y t y t f t dt y x y x y x f x
y x y x y x y x y x y x
= +

Now we take the second derivative
1 2 3
(2) (1) (1) (1)
0 1 2 3
(2) (2) (2)
0
1 2 3
( ) ( ) ( )
( ) ( ) ( ) ( ) ( )
( ) ( ) ( )
x
y t y t y t
y x y t y t y t f t dt
y x y x y x
= 

Now, in the same way, we get the derivative of the third order
1 2 3 1 2 3
(3) (1) (1) (1) (1) (1) (1)
0 1 2 3 1 2 3
(3) (3) (3) (2) (2) (2)
0
1 2 3 1 2 3
( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) ( )
x
y t y t y t y x y x y x
y x y t y t y t f t dt y x y x y x f x
y x y x y x y x y x y x
= +

Now, for the correctness of the received function, we must establish and check the price of the function and its
derivatives of different degrees in equation (1).
(3) (1)
0 1 0 0 0
( ) ( ) ( ) ( ) ( ) ( )
L y y x P x y x P x y x
= + +
1 2 3
(1) (1) (1)
1 2 3
(3) (3) (3)
0
1 2 3
1 2 3
(1) (1) (1)
1 1 2 3
(1) (1) (1)
0
1 2 3
1 2 3
(1) (1)
0 1 2 3
( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( )
( ) ( ) ( )
( ) ( ) ( ) ( ) ( )
( ) ( ) ( )
( ) ( ) ( )
( ) ( ) ( )
x
x
y t y t y t
L y y t y t y t f t dt f x
y x y x y x
y t y t y t
P x y t y t y t f t dt
y x y x y x
y t y t y t
P x y t y t y
= + +
+
+


(1)
0
1 2 3
( ) ( )
( ) ( ) ( )
x
t f t dt
y x y x y x

From here we add the determinants together,
( )
L y =
1 2 3
(1) (1) (1)
1 2 3
(3) (1) (3) (1) (3) (1)
0
1 1 1 0 1 2 1 2 0 2 3 1 3 0 3
( ) ( ) ( )
( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
x
y t y t y t
y t y t y t
y x P x y x P x y x y x P x y x P x y x y x P x y x P x y x
=
+ + + + + +

( ) ( ).
f t dt f x
+
Conditions of homogeneous equation (3) (1)
1 1 1 0 1
( ) ( ) ( ) ( ) ( ) ( ) 0
L y y x P x y x P x y x
= + + = so that we can solvefunction
of ( )
f x and as a result we can say that we have obtained the solution of the inhomogeneous part.
We have obtained the Green's function for the proposed problem and according to the problem, we have proved
that:
(5)
( ) ( ), 0 1
L y f x x
= < <
with boundary conditions
1 2 3
( ) 0 , ( ) 0, ( ) 0 .
U y U y U y
= = = (6)
The type of boundary conditions that are already defined for us.
1 1 1
( ) (0) (1) 0
U y y y
α β
= − =
2 2 2
( ) (0) (1) 0
U y y y
α β
′ ′
= − =
3 3 3
( ) (0) (1) 0
U y y y
α β
′′ ′′
= − =
It can be said that we can solve the equation and Green's function (5), (6) using Green's functions as follows.
1
1
0 0
0
( , ) ( ) ( , , ) ( )
y x t L I f G x t f t dt
λ λ
−
= − = 

where
1 2 3
1 1 1 2 1 3 1
2 1 2 2 2 3 2
3 1 3 2 3 3 3
0
1 1 1 2 1 3
2 1 2 2 2 3
3 1 3 2 3 3
( , ) ( , ) ( , ) ( , )
( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
( , , )
(y ) (y ) (y )
(y ) (y ) (y )
(y ) (y ) (y )
y x y x y x g x t
U y U y U y U g
U y U y U y U g
U y U y U y U g
G x t
U U U
U U U
U U U
λ λ λ
λ = −
0 ( , , )
G x t λ − is a Green's function.
we assume
2 1 0
3 0
γ γ γ
≥ ≥ ≥ ≥
4. Main results
1 2 3
1 1 1 2 1 3 1
2 1 2 2 2 3 2
1
3 1 3 2 3 3 3
1
0
1 1 1 2 1 3
0
2 1 2 2 2 3
3 1 3 2 3 3
( , ) ( , ) ( , ) ( , )
( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
( , ) ( ) ( )
(y ) (y ) (y )
(y ) (y ) (y )
(y ) (y ) (y )
y x y x y x g x t
U y U y U y U g
U y U y U y U g
U y U y U y U g
y x t L I f f t dt
U U U
U U U
U U U
λ λ λ
λ −
= − = −
If t
x > the function )
,
( t
x
g has the following form
1 2 3
(1) (1) (1)
1 2 3
1 1 1
( ) ( ) ( )
( , ) ( ) ( ) ( )
( ) ( ) ( )
y t y t y t
g x t y t y t y t
y x y x y x
=
If x t
≤ is then a function 0
)
,
( =
t
x
g .
1 1 1 2 1 3
0 2 1 2 2 2 3
3 1 3 2 3 3
(y ) (y ) (y )
( ) (y ) (y ) (y )
(y ) (y ) (y )
U U U
U U U
U U U
λ
∆ =
5. Discussion
From the topic of research, we come to the conclusion
that the problem we studied in inhomogeneous linear
differential equations of the third order is a set of
Green's function. Every real function exists in the
solution of a set of linear differential equations, and
such equations have not only one definite solution but
also several solutions. Its field of application in
physics, for example, finding the temperature at all
points of an iron rod with one end at absolute zero
and the other end at the freezing point of water, is a
boundary value problem.
If the problem depends on both space and time, the
value of the problem can be determined at a certain
point for all times or at a certain time for the entire
space and provide another example of a linear
differential equation with boundary conditions.
The boundary condition that determines the value of
the function is the Dirichlet boundary condition. For
example, if one end of an iron rod is held at absolute
zero, the magnitude of the problem is determined at
that point in space.
6. Conclusion
Since we have obtained the Green's function for
solving the third-order inhomogeneous linear
differential equation, everything in this system is
technically solvable. To solve it, we proposed the
received method and showed that the inhomogeneous
linear differential equation of the third order with the
boundary conditions of the problem does not have
one solution, but has several solutions in terms of
eigenvalues and eigenfunctions.

References
[1] Kanguzhin, B., Aimal Rasa, G.H. and
Kaiyrbek, Z., 2021. Identification of the
domain of the sturm–liouville operator on a star
graph. Symmetry, 13(7), p.1210.
[2] Rasa, G.H.A. and Auzerkhan, G., 2021.
INCEPTION OF GREEN FUNCTION FOR
THE THIRD-ORDER LINEAR
DIFFERENTIAL EQUATION THAT IS
INCONSISTENT WITH THE BOUNDARY
PROBLEM CONDITIONS. Journal of
Mathematics, Mechanics & Computer
Science, 110(2).
[3] Rasa, G.H.A., 2020. The Analytical Nature of
the Green's Function in the Vicinity of a Simple
Pole. International Journal of Trend in
Scientific Research and Development (IJTSRD)
Volume, 4.
[4] Kanguzhin, B.E., Dairbayeva, G. and
Madibaiuly, Z., 2019. Единственность
восстановления граничных условий
дифференциального оператора по набору
спектров. Journal of Mathematics, Mechanics
and Computer Science, 104(4), pp.44-49.
[5] Keselman G.M. "Bezuslovnaya shodimost
razlozheniy po sobstvennyim funktsiyam
nekotoryih diﬀerentsialnyih operatorov
[Unconditional convergence of expansions in
eigenfunctions of some diﬀerential operators]",
Izv. Universitetyi SSSR, ser. mat. 2 (2017):82-
93.
[6] Kanguzhin, B.E. and Aniyarov, A.A., 2011.
Well-posed problems for the Laplace operator
in a punctured disk. Mathematical Notes, 89,
pp.819-829.
[7] Rasa, G.H.A., 2022. Formulas for Surface
Weighted Numbers on Graph.
[8] Михайлов В.П. О базисах Рисса // ДАН
СССР.–2018.№ 5 (144).–С. 981-984.
[9] Кесельман, Г.М., 1964. О безусловной
сходимости разложений по собственным
функциям некоторых дифференциальных
операторов. Известия высших учебных
заведений. Математика, (2), pp.82-93.
[10] Наймарк, М.А., 2010. Линейные
дифференциальные операторы. Физматлит.
pp 528.
[11] Ras, G.H.A., 2021. Asymptotic Formulas for
Weight Numbers of the Boundary Problem
differential operator on a Star-shaped
Graph. Turkish Journal of Computer and
Mathematics Education (TURCOMAT), 12(13),
pp.2184-2192.
[12] Rasa, G.H.A., 2023. Residual Decomposition
of the Green's Function of the Dirichlet
problem for a Differential Operator on a star-
graph for m= 2. Turkish Journal of Computer
and Mathematics Education
(TURCOMAT), 14(2), pp.132-140.
[13] Nasri, F. and Rasa, G.H.A., 2024. Lagrange
formula conjugate third order differential
equation. Turkish Journal of Computer and
Mathematics Education (TURCOMAT), 15(1),
pp.70-74.
[14] Rasa GR, Rasa GH. Sturm-Liouville problem
with general inverse symmetric potential.
Science and Education. 2023;4(8):7-15.

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