scholarly journals Analytic solution of transcendental equations

2010 ◽  
Vol 20 (4) ◽  
pp. 671-677 ◽  
Author(s):  
Henryk Górecki
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Linear Differential Equation ◽  
Analytic Solution ◽  
Linear Differential Equations ◽  
Order Linear Differential Equation ◽  
Decomposition Technique ◽  
Order Linear ◽  
Transcendental Equations ◽  
Linear Differential

Analytic solution of transcendental equationsA decomposition technique of the solution of ann-th order linear differential equation into a set of solutions of 2-nd order linear differential equations is presented.

scholarly journals Oscillation of first order linear differential equations with several non-monotone delays

2018 ◽  
Vol 16 (1) ◽  
pp. 83-94
Author(s):  
E.R. Attia ◽  
V. Benekas ◽  
H.A. El-Morshedy ◽  
I.P. Stavroulakis
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Linear Differential Equation ◽  
Linear Differential Equations ◽  
Order Linear Differential Equation ◽  
Order Linear ◽  
First Order ◽  
Linear Differential

AbstractConsider the first-order linear differential equation with several retarded arguments$$\begin{array}{} \displaystyle x^{\prime }(t)+\sum\limits_{k=1}^{n}p_{k}(t)x(\tau _{k}(t))=0,\;\;\;t\geq t_{0}, \end{array} $$where the functions pk, τk ∈ C([t0, ∞), ℝ+), τk(t) < t for t ≥ t0 and limt→∞τk(t) = ∞, for every k = 1, 2, …, n. Oscillation conditions which essentially improve known results in the literature are established. An example illustrating the results is given.

scholarly journals Generalized Hyers-Ulam Stability of the Second-Order Linear Differential Equations

2011 ◽  
Vol 2011 ◽  
pp. 1-10 ◽  
Author(s):  
A. Javadian ◽  
E. Sorouri ◽  
G. H. Kim ◽  
M. Eshaghi Gordji
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Linear Differential Equation ◽  
Open Interval ◽  
Second Order ◽  
Linear Differential Equations ◽  
Ulam Stability ◽  
Order Linear Differential Equation ◽  
Order Linear ◽  
Linear Differential

We prove the generalized Hyers-Ulam stability of the 2nd-order linear differential equation of the form , with condition that there exists a nonzero in such that and is an open interval. As a consequence of our main theorem, we prove the generalized Hyers-Ulam stability of several important well-known differential equations.

scholarly journals ELEMENTARY METHOD OF SUBSTANTIATION AND IDEOLOGICAL MEANING OF THE THEORY OF THE SECOND ORDER LINEAR DIFFERENTIAL EQUATIONS WITH CONSTANT COEFFICIENTS AND TRIGONOMETRIC FREE TERMS

2021 ◽  
pp. 105-114
Author(s):  
Zh. A. Sartabanov ◽  
A. Kh. Zhumagaziyev ◽  
A. A. Duyussova
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Linear Differential Equation ◽  
Teaching Method ◽  
Second Order ◽  
School Mathematics ◽  
Linear Differential Equations ◽  
Order Linear ◽  
Constant Coefficients ◽  
Linear Differential

In the article, adapted to the school course, the second order linear differential equations with constant coefficients and trigonometric free terms are investigated. The basic elementary methodological approaches to solving the equation are given. The solutions of the second order linear differential equation with constant coefficients and trigonometric free terms are investigated, which is a model of many phenomena. In addition, the applied values of the equation and its solutions were noted. The results obtained are presented in the form of theorems. The main novelty of the study is that these results are proved and generalized by elementary methods. These conclusions are proved in the framework of the methods of high school mathematics. This theory, known in general mathematics, is fully adapted to the implementation in secondary school mathematics and developed with the help of new elementary techniques that are understandable to the student. The main purpose of the research is to develop methods for solving a non-uniform linear differential equation of the second order with a constant coefficient at a level that a schoolboy can master. The result will be the creation of a special course program on the basics of ordinary differential equations in secondary schools of the natural-mathematical direction, the preparation of appropriate content material and providing them with a simple teaching method.

SOME PROPERTIES OF COMBINATION OF SOLUTIONS TO SECOND-ORDER LINEAR DIFFERENTIAL EQUATIONS WITH ANALYTIC COEFFICIENTS OF [P;Q]-ORDER IN THE UNIT DISC

2015 ◽  
Vol 1 (1) ◽  
pp. 11-18
Author(s):  
Benharrat Belaïdi ◽  
Zinelâabidine Latreuch
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Linear Differential Equation ◽  
Unit Disc ◽  
Second Order ◽  
Linear Differential Equations ◽  
Order Linear ◽  
Linear Differential

In this paper, we consider some properties on the growth and oscillation of combination of solutions of the linear differential equation \[f'' + A(z) f' + B (z) f = 0\] with analytic coefficients A(z) and B (z) with [p; q]-order in the unit disc $\Delta = \{z \in \mathbb{C} : |z| < 1\}$.

Oscillation of first-order differential equations with several non-monotone retarded arguments

2020 ◽  
Vol 27 (3) ◽  
pp. 341-350 ◽  
Author(s):  
Huseyin Bereketoglu ◽  
Fatma Karakoc ◽  
Gizem S. Oztepe ◽  
Ioannis P. Stavroulakis
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Linear Differential Equation ◽  
Order Linear Differential Equation ◽  
Oscillation Criteria ◽  
Order Linear ◽  
First Order ◽  
Linear Differential ◽  
First Order Differential Equations

AbstractConsider the first-order linear differential equation with several non-monotone retarded arguments {x^{\prime}(t)+\sum_{i=1}^{m}p_{i}(t)x(\tau_{i}(t))=0}, {t\geq t_{0}}, where the functions {p_{i},\tau_{i}\in C([t_{0},\infty),\mathbb{R}^{+})}, for every {i=1,2,\ldots,m}, {\tau_{i}(t)\leq t} for {t\geq t_{0}} and {\lim_{t\to\infty}\tau_{i}(t)=\infty}. New oscillation criteria which essentially improve the known results in the literature are established. An example illustrating the results is given.

scholarly journals Some Properties of Solutions of Second-Order Linear Differential Equations

2013 ◽  
Vol 2013 ◽  
pp. 1-5
Author(s):  
Zinelaâbidine Latreuch ◽  
Benharrat Belaïdi
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Linear Differential Equation ◽  
Finite Order ◽  
Entire Functions ◽  
Second Order ◽  
Linear Differential Equations ◽  
Order Linear ◽  
Linearly Independent ◽  
Linear Differential

We study the growth and oscillation of gf=d1f1+d2f2, where d1 and d2 are entire functions of finite order not all vanishing identically and f1 and f2 are two linearly independent solutions of the linear differential equation f′′+A(z)f=0.

scholarly journals Determining a Particular Solution for n-th Order Linear Differential Equations with Constant Coefficients

2017 ◽  
Vol 23 (3) ◽  
pp. 24-29
Author(s):  
Vasile Căruțașu
Keyword(s):  
Differential Equation ◽  
General Solution ◽  
Differential Equations ◽  
Linear Differential Equation ◽  
Linear Differential Equations ◽  
Free Term ◽  
Order Linear ◽  
Constant Coefficients ◽  
Particular Solution ◽  
Linear Differential

Abstract For n-th order linear differential equations with constant coefficients, the problem to be solved is related to determining a particular solution, and then, with the general solution of n-th homogeneous linear differential equation with constant coefficients attached, to write the general solution of n-th linear differential equation with the given constant coefficients. In all the works that deal with this issue three situations are analyzed: the situation in which the free term is a polynomial P(x), the situation in which the free term is like P(x)· eα·x and lastly, the situation in which the free term is like eω·x · (P(x)· cos(β·x)+ Q(x)·sin(β·x)). In this study we aim to analyze if the free term is a combination of the three cases mentioned.

scholarly journals Solutions ofk-Hypergeometric Differential Equations

2014 ◽  
Vol 2014 ◽  
pp. 1-13 ◽  
Author(s):  
Shahid Mubeen ◽  
Mammona Naz ◽  
Abdur Rehman ◽  
Gauhar Rahman
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Linear Differential Equation ◽  
Second Order ◽  
Order Linear Differential Equation ◽  
Frobenius Method ◽  
Order Linear ◽  
Hypergeometric Differential Equation ◽  
Linear Differential

We solve the second-order linear differential equation called thek-hypergeometric differential equation by using Frobenius method around all its regular singularities. At each singularity, we find 8 solutions corresponding to the different cases for parameters and modified our solutions accordingly.

Properties 𝐴 and 𝐵 of 𝑛th Order Linear Differential Equations with Deviating Argument

1999 ◽  
Vol 6 (6) ◽  
pp. 553-566
Author(s):  
R. Koplatadze ◽  
G. Kvinikadze ◽  
I. P. Stavroulakis
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Linear Differential Equation ◽  
Ordinary Differential Equations ◽  
Sufficient Conditions ◽  
Order Linear Differential Equation ◽  
Deviating Arguments ◽  
Order Linear ◽  
Deviating Argument ◽  
Linear Differential

Abstract Sufficient conditions for the nth order linear differential equation 𝑢(𝑛) (𝑡) + 𝑝(𝑡)𝑢(τ(𝑡)) = 0, 𝑛 ≥ 2, to have Property 𝐴 or Property 𝐵 are established in both the delayed and the advanced cases. These conditions essentially improve many known results not only for differential equations with deviating arguments but for ordinary differential equations as well.

On a Relation Between a Theorem of Hartman and a Theorem of Sherman

1973 ◽  
Vol 16 (2) ◽  
pp. 275-281 ◽  
Author(s):  
A. C. Peterson
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Linear Differential Equation ◽  
General Result ◽  
Nonlinear Differential Equations ◽  
Conjugate Point ◽  
Order Linear Differential Equation ◽  
Order Linear ◽  
Linear Differential ◽  
Image Position

We are concerned with the nth-order linear differential equation1where the coefficients are assumed to be continuous. Hartman [1] proved that (see Definition 2) the first conjugate point η1(t) of t satisfies2Hartman actually proved a more general result which has very important applications in nonlinear differential equations.

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