scholarly journals Some comparison criteria in oscillation theory

1984 ◽  
Vol 36 (2) ◽  
pp. 176-186 ◽  
Author(s):  
Ch. G. Philos
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Asymptotic Behavior ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Arbitrary Order ◽  
Oscillation Theory ◽  
Linear Ordinary Differential Equation ◽  
Order Linear ◽  
Comparison Criteria

AbstractThe purpose of this paper is to establish comparison criteria, by which the oscillatory and asymptotic behavior of linear retarded differential equations of arbitrary order is inherited from the oscillation of an associated second order linear ordinary differential equation. These criteria are new even in the case of ordinary differential equations.

scholarly journals Positive decreasing solutions of second order quasilinear ordinary differential equations in the framework of regular variation

Filomat ◽  
2015 ◽  
Vol 29 (9) ◽  
pp. 1995-2010 ◽  
Author(s):  
Jelena Milosevic ◽  
Jelena Manojlovic
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Asymptotic Behavior ◽  
Differential Equations ◽  
Asymptotic Analysis ◽  
Ordinary Differential Equations ◽  
Regular Variation ◽  
Second Order ◽  
Precise Information ◽  
Varying Coefficients

This paper is concerned with asymptotic analysis of positive decreasing solutions of the secondorder quasilinear ordinary differential equation (E) (p(t)?(|x'(t)|))'=q(t)?(x(t)), with the regularly varying coefficients p, q, ?, ?. An application of the theory of regular variation gives the possibility of determining the precise information about asymptotic behavior at infinity of solutions of equation (E) such that lim t?? x(t)=0, lim t?? p(t)?(-x'(t))=?.

Qualitative properties of nonlinear ordinary differential equations

1977 ◽  
Vol 79 (1-2) ◽  
pp. 79-85 ◽  
Author(s):  
Nelson Onuchic ◽  
Plácido Z. Táboas
Keyword(s):  
Differential Equation ◽  
Banach Space ◽  
Ordinary Differential Equation ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Linear Ordinary Differential Equation ◽  
Point Of View ◽  
Nonlinear Ordinary Differential Equations ◽  
Qualitative Properties ◽  
Image Position

SynopsisThe perturbed linear ordinary differential equationis considered. Adopting the same approach of Massera and Schäffer [6], Corduneanu states in [2] the existence of a set of solutions of (1) contained in a given Banach space. In this paper we investigate some topological aspects of the set and analyze some of the implications from a point of view ofstability theory.

Asymptotic formulas for solutions of a singular linear ordinary differential equation

1978 ◽  
Vol 81 (1-2) ◽  
pp. 57-70 ◽  
Author(s):  
Richard C. Gilbert
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Differential Operator ◽  
Differential Equations ◽  
Asymptotic Expansions ◽  
Linear Ordinary Differential Equation ◽  
Asymptotic Formulas ◽  
Order Linear ◽  
First Order ◽  
Complex Coefficients

SynopsisBy use of the theory of asymptotic expansions for first-order linear systems of ordinary differential equations, asymptotic formulas are obtained for the solutions of annth order linear homogeneous ordinary differential equation with complex coefficients having asymptotic expansions in a sector of the complex plane. These asymptotic formulas involve the roots of certain polynomials whose coefficients are obtained from the asymptotic expansions of the coefficients of the differential operator.

scholarly journals Hidden and Not So Hidden Symmetries

2012 ◽  
Vol 2012 ◽  
pp. 1-11 ◽  
Author(s):  
P. G. L. Leach ◽  
K. S. Govinder ◽  
K. Andriopoulos
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Recent Work ◽  
Nonlocal Symmetry ◽  
Partial Differential ◽  
Hidden Symmetries ◽  
Contact Symmetry

Hidden symmetries entered the literature in the late Eighties when it was observed that there could be gain of Lie point symmetry in the reduction of order of an ordinary differential equation. Subsequently the reverse process was also observed. Such symmetries were termed “hidden”. In each case the source of the “new” symmetry was a contact symmetry or a nonlocal symmetry, that is, a symmetry with one or more of the coefficient functions containing an integral. Recent work by Abraham-Shrauner and Govinder (2006) on the reduction of partial differential equations demonstrates that it is possible for these “hidden” symmetries to have a point origin. In this paper we show that the same phenomenon can be observed in the reduction of ordinary differential equations and in a sense loosen the interpretation of hidden symmetries.

scholarly journals On classical symmetries of ordinary differential equations related to stationary integrable partial differential equations

2021 ◽  
Vol 41 (5) ◽  
pp. 685-699
Author(s):  
Ivan Tsyfra
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Theoretical Method ◽  
Partial Differential ◽  
Classical Symmetry ◽  
Kdv Equations ◽  
Classical Symmetries

We study the relationship between the solutions of stationary integrable partial and ordinary differential equations and coefficients of the second-order ordinary differential equations invariant with respect to one-parameter Lie group. The classical symmetry method is applied. We prove that if the coefficients of ordinary differential equation satisfy the stationary integrable partial differential equation with two independent variables then the ordinary differential equation is integrable by quadratures. If special solutions of integrable partial differential equations are chosen then the coefficients satisfy the stationary KdV equations. It was shown that the Ermakov equation belong to a class of these equations. In the framework of the approach we obtained the similar results for generalized Riccati equations. By using operator of invariant differentiation we describe a class of higher order ordinary differential equations for which the group-theoretical method enables us to reduce the order of ordinary differential equation.

scholarly journals On Conjugacy of High-Order Linear Ordinary Differential Equations

1994 ◽  
Vol 1 (1) ◽  
pp. 1-8
Author(s):  
T. Chanturia
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
High Order ◽  
Summable Function ◽  
Order Linear ◽  
Linear Ordinary Differential Equations

Abstract It is shown that the differential equation u (n) = p(t)u, where n ≥ 2 and p : [a, b] → ℝ is a summable function, is not conjugate in the segment [a, b], if for some l ∈ {1, . . . , n – 1}, α ∈]a, b[ and β ∈]α, b[ the inequalities hold.

DERIVATION OF 2-POINT ZERO STABLENUMERICAL ALGORITHM OF BLOCK BACKWARD DIFFERENTIATION FORMULA FOR SOLVING FIRST ORDER ORDINARY DIFFERENTIAL EQUATIONS

2021 ◽  
Vol 5 (2) ◽  
pp. 579-583
Author(s):  
Muhammad Abdullahi ◽  
Bashir Sule ◽  
Mustapha Isyaku
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Taylor Series ◽  
Taylor Series Expansion ◽  
Order Ordinary Differential Equation ◽  
Differentiation Formula ◽  
First Order ◽  
Backward Differentiation Formula

This paper is aimed at deriving a 2-point zero stable numerical algorithm of block backward differentiation formula using Taylor series expansion, for solving first order ordinary differential equation. The order and zero stability of the method are investigated and the derived method is found to be zero stable and of order 3. Hence, the method is suitable for solving first order ordinary differential equation. Implementation of the method has been considered

Stability of Solutions of Ordinary Differential Equations with Respect to a Closed Set

1968 ◽  
Vol 20 ◽  
pp. 720-726
Author(s):  
T. G. Hallam ◽  
V. Komkov
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Compact Set ◽  
Stability Of Solutions ◽  
Closed Set ◽  
The Stability ◽  
Stability Results

The stability of the solutions of an ordinary differential equation will be discussed here. The purpose of this note is to compare the stability results which are valid with respect to a compact set and the stability results valid with respect to an unbounded set. The stability of sets is a generalization of stability in the sense of Liapunov and has been discussed by LaSalle (5; 6), LaSalle and Lefschetz (7, p. 58), and Yoshizawa (8; 9; 10).

Construction of Green’s functional for a third order ordinary differential equation with general nonlocal conditions and variable principal coefficient

2020 ◽  
Vol 27 (4) ◽  
pp. 593-603 ◽  
Author(s):  
Kemal Özen
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Nonlocal Problem ◽  
Nonlocal Conditions ◽  
Adjoint Problem ◽  
Linear Ordinary Differential Equation ◽  
Third Order ◽  
Order Ordinary Differential Equation ◽  
Order Linear ◽  
Theoretical Results

AbstractIn this work, the solvability of a generally nonlocal problem is investigated for a third order linear ordinary differential equation with variable principal coefficient. A novel adjoint problem and Green’s functional are constructed for a completely nonhomogeneous problem. Several illustrative applications for the theoretical results are provided.

scholarly journals On the Polynomial Solutions and Limit Cycles of Some Generalized Polynomial Ordinary Differential Equations

Mathematics ◽  
2020 ◽  
Vol 8 (7) ◽  
pp. 1139
Author(s):  
Claudia Valls
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Limit Cycles ◽  
Polynomial Solutions ◽  
Generalized Polynomial

We study equations of the form y d y / d x = P ( x , y ) where P ( x , y ) ∈ R [ x , y ] with degree n in the y-variable. We prove that this ordinary differential equation has at most n polynomial solutions (not necessarily constant but coprime among each other) and this bound is sharp. We also consider polynomial limit cycles and their multiplicity.

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