On Constants in Nonoscillation Criteria for Half-Linear Differential Equations

Riccati Technique ◽

Linear Differential ◽

Parametric Expression

We study the half-linear differential equation(r(t)Φ(x′))′+c(t)Φ(x)=0, whereΦ(x)=|x|p−2x,p>1. Using the modified Riccati technique, we derive new nonoscillation criteria for this equation. The results are closely related to the classical Hille-Nehari criteria and allow to replace the fixed constants in known nonoscillation criteria by a certain one-parametric expression.

Use of the Modified Riccati Technique for Neutral Half-Linear Differential Equations

Mathematics ◽

10.3390/math9030235 ◽

2021 ◽

Vol 9 (3) ◽

pp. 235

Author(s):

Zuzana Pátíková ◽

Simona Fišnarová

Keyword(s):

Differential Equation ◽

Differential Equations ◽

Type Equation ◽

Second Order ◽

Riccati Technique ◽

Oscillation Criteria ◽

Type Criterion ◽

We study the second-order neutral half-linear differential equation and formulate new oscillation criteria for this equation, which are obtained through the use of the modified Riccati technique. In the first statement, the oscillation of the equation is ensured by the divergence of a certain integral. The second one provides the condition of the oscillation in the case where the relevant integral converges, and it can be seen as a Hille–Nehari-type criterion. The use of the results is shown in several examples, in which the Euler-type equation and its perturbations are considered.

On [p,q]-order of growth of solutions of complex linear differential equations near a singular point

Filomat ◽

10.2298/fil1913013l ◽

2019 ◽

Vol 33 (13) ◽

pp. 4013-4020

Author(s):

Jianren Long ◽

Sangui Zeng

Keyword(s):

Differential Equation ◽

Singular Point ◽

Differential Equations ◽

Order Of Growth ◽

Complex Linear ◽

Linear Differential ◽

Growth Of Solutions

We investigate the [p,q]-order of growth of solutions of the following complex linear differential equation f(k)+Ak-1(z) f(k-1) + ...+ A1(z) f? + A0(z) f = 0, where Aj(z) are analytic in C? - {z0}, z0 ? C. Some estimations of [p,q]-order of growth of solutions of the equation are obtained, which is generalization of previous results from Fettouch-Hamouda.

Relation between Small Functions with Differential Polynomials Generated by Solutions of Linear Differential Equations

Abstract and Applied Analysis ◽

10.1155/2012/451825 ◽

2012 ◽

Vol 2012 ◽

pp. 1-11

Author(s):

Zhigang Huang

Keyword(s):

Differential Equation ◽

Differential Equations ◽

Entire Functions ◽

Differential Polynomials ◽

Small Functions ◽

Linear Differential ◽

The Relationship ◽

Growth Of Solutions

This paper is devoted to studying the growth of solutions of second-order nonhomogeneous linear differential equation with meromorphic coefficients. We also discuss the relationship between small functions and differential polynomialsL(f)=d2f″+d1f′+d0fgenerated by solutions of the above equation, whered0(z),d1(z),andd2(z)are entire functions that are not all equal to zero.

Studies in multiplicative solutions to linear differential equations

Proceedings of the Royal Society of Edinburgh Section A Mathematics ◽

10.1017/s0308210500018424 ◽

1987 ◽

Vol 106 (3-4) ◽

pp. 277-305 ◽

Cited By ~ 2

Author(s):

F. M. Arscott

Keyword(s):

Differential Equation ◽

Differential Equations ◽

Floquet Theory ◽

Special Functions ◽

Ordinary Linear Differential Equation ◽

Closed Path ◽

Starting Point ◽

SynopsisGiven an ordinary linear differential equation whose singularities are isolated, a solution is called multiplicative for a closed path C if, when continued analytically along C, it returns to its starting-point merely multiplied by a constant. This paper first classifies such paths into three types, then investigates combinations of two such paths, in which a number of qualitatively different situations can arise. A key result is also given relating to a three-path combination. There are applications to special functions and Floquet theory for periodic equations.

So-Called Cases of Failure in the Solution of Linear Differential Equations

The Mathematical Gazette ◽

10.1017/s0025557200245439 ◽

1916 ◽

Vol 8 (123) ◽

pp. 258-262

Author(s):

Eric H. Neville

Keyword(s):

Differential Equation ◽

General Solution ◽

Differential Equations ◽

General Equation ◽

Effective Constant ◽

Linear Differential ◽

Made In

There are two ways in which the solution of a particular linear differential equation may “fail” although the solulion of a more general equation obtained by replacing certain constants by parameters is complete.where D as usual stands for d/dx.For the general equation(D — l)(D — m)y = enxthe perfectly general solution isA, B being independent arbitrary constants, but if we attempt to apply this solution to the particular equation (l), we find in the first place that the coincidence of n with l and m renders the first term infinite, and in the second place that the coincidence of m with l leaves us with only one effective constant, A + B. The method by which in the commoner textbooks the passage from the general solution to that of a particular equation is made in such cases as this is unconvincing.

New formula for solution of one class of linear differential equations of the second order with the variable coefficients

Transactions on Machine Learning and Artificial Intelligence ◽

10.14738/tmlai.83.8402 ◽

2020 ◽

Vol 8 (3) ◽

pp. 61-68

Author(s):

Avyt Asanov ◽

Kanykei Asanova

Keyword(s):

Differential Equation ◽

Differential Equations ◽

Variable Coefficients ◽

Second Order ◽

Common Solution ◽

The Common ◽

Linear Differential ◽

New Formula

Exact solutions for linear and nonlinear differential equations play an important rolein theoretical and practical research. In particular many works have been devoted tofinding a formula for solving second order linear differential equations with variablecoefficients. In this paper we obtained the formula for the common solution of thelinear differential equation of the second order with the variable coefficients in themore common case. We also obtained the new formula for the solution of the Cauchyproblem for the linear differential equation of the second order with the variablecoefficients.Examples illustrating the application of the obtained formula for solvingsecond-order linear differential equations are given.Key words: The linear differential equation, the second order, the variablecoefficients,the new formula for the common solution, Cauchy problem, examples.

IV. On linear differential equations

Proceedings of the Royal Society of London ◽

10.1098/rspl.1869.0031 ◽

1870 ◽

Vol 18 (114-122) ◽

pp. 118-119

Keyword(s):

Differential Equation ◽

Differential Equations ◽

Rational Function ◽

System Of Equations ◽

The condition that the linear differential equation (α + β x + γ x 2 ) d 2 u / dx 2 + (α' + β' x + γ' x 2 ) du / dx + (α'' + β" x + γ" x 2 ) u = 0 admits of an integral u = ϵ fφdx , where φ is a rational function of ( x ), is given by the system of equations

X. On linear differential equations.—No. V

Proceedings of the Royal Society of London ◽

10.1098/rspl.1870.0089 ◽

1871 ◽

Vol 19 (123-129) ◽

pp. 526-528

Keyword(s):

Differential Equation ◽

Differential Equations ◽

Rational Functions ◽

Let us now endeavour to ascertain under what circumstance a linear differential equation admits a solution of the form P log e Q, where P and Q are rational functions of ( x ).

Pseudo-autonomous linear systems

Bulletin of the Australian Mathematical Society ◽

10.1017/s0004972700023005 ◽

1977 ◽

Vol 16 (1) ◽

pp. 61-65 ◽

Cited By ~ 8

Author(s):

W.A. Coppel

Keyword(s):

Differential Equation ◽

Differential Equations ◽

Linear Systems ◽

Point Spectrum ◽

Coefficient Matrix ◽

Pure Point ◽

Pure Point Spectrum ◽

Pseudo-autonomous linear differential equations are defined. A linear differential equation with bounded coefficient matrix is pseudo-autonomous if and only if it is almost reducible. A linear differential equation with recurrent coefficient matrix is pseudo-autonomous if and only if it has pure point spectrum.

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

Open Mathematics ◽

10.1515/math-2018-0010 ◽

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 ◽

Order Linear Differential Equation ◽

Order Linear ◽

First Order ◽

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.