scholarly journals Some Study Methods for Ordinary Differential Equation Integrability of the Second Order of a Certain Type

2019 ◽  
pp. 48-62
Author(s):  
O. V. Zadorozhnaya ◽  
V. K. Kochetkov
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Nonlinear Differential Equation ◽  
Nonlinear Differential Equations ◽  
Practical Interest ◽  
Second Order ◽  
Parameter Method ◽  
Study Methods ◽  
Special Cases ◽  
General Conditions

The paper deals with treating some study methods of the equation integrability of a certain type that are little studied in the theory of differential equations. It is known that a significant part of the differential equations cannot be integrated. Then, to develop methods for their study is, certainly, of scientific interest. The obtained results, formulated as theorems and statements, are of scientific and practical interest because of their importance for applications in modern science.In the paper we present an alternative method for studying the integrability of both linear and nonlinear differential equations of the second order. An introduction of parameters allowed us to develop a study method for the integrability of ordinary differential equations of the second order. We also formulate the theorems describing some General conditions for the integrability of the second-order linear equation and consider special cases of integrability, which arise out of the above facts.Based on the obtained parameter method, some General conditions for the integrability of the nonlinear differential equation of the second order are given, and the consequences of these General conditions are indicated.We have obtained new results related to the construction and development of methods for studying the differential equation to which some types of differential equations are reduced and laid the foundations for a rigorous and systematic study of the introduced special nonlinear differential equation of the second order.

Oscillation of Second-Order Nonlinear Differential Equations

2014 ◽  
Vol 548-549 ◽  
pp. 1007-1010
Author(s):  
Qing Zhu ◽  
Zhi Bin Ma
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Nonlinear Differential Equation ◽  
Nonlinear Differential Equations ◽  
Second Order ◽  
Oscillation Criterion ◽  
Order Nonlinear Differential Equation

A new oscillation criterion is established for a certain class of second-order nonlinear differential equation x"(t)-b(t)x'(t)+c(t)g(x)=0, x"(t)+c(t)g(x)=0 that is different from most known ones. Some applications of the result obtained are also presented. Our results are sharper than some previous ones.

scholarly journals Oscillation and nonoscillation theorems for second order nonlinear differential equations

2001 ◽  
Vol 32 (2) ◽  
pp. 95-102
Author(s):  
Jiang Jianchu
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Nonlinear Differential Equation ◽  
Nonlinear Differential Equations ◽  
Second Order ◽  
Order Nonlinear Differential Equation ◽  
Oscillation And Nonoscillation

New oscillation and nonoscillation theorems are obtained for the second order nonlinear differential equation $$ (|u'(t)|^{\alpha -1} u'(t))' + p(t)|u(t)|^{\alpha -1} u(t) = 0 $$ where $ p(t) \in C [0, \infty) $ and $ p(t) \ge 0 $. Conditions only about the integrals of $ p(t) $ on every interval $ [2^n t_0, 2^{n+1} t_0] $ ($ n = 1, 2, \ldots $) for some fixed $ t_0 >0 $ are used in the results.

Oscillation Criteria for Second Order Nonlinear Differential Equations Involving Integral Averages

1993 ◽  
Vol 45 (5) ◽  
pp. 1094-1103 ◽  
Author(s):  
James S. W. Wong
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Linear Equation ◽  
Nonlinear Differential Equation ◽  
General Equation ◽  
Nonlinear Differential Equations ◽  
Nonlinear Function ◽  
Second Order ◽  
Oscillation Criteria ◽  
Fowler Equation

AbstractConsider the second order nonlinear differential equationy" + a(t)f(y) = 0where a(t) ∈ C[0,∞),f(y) GC1 (-∞, ∞),ƒ'(y) ≥ 0 and yf(y) > 0 for y ≠ 0. Furthermore, f(y) also satisfies either a superlinear or a sublinear condition, which covers the prototype nonlinear function f(y) = |γ|γ sgny with γ > 1 and 0 < γ < 1 known as the Emden-Fowler case. The coefficient a(t) is allowed to be negative for arbitrarily large values of t. Oscillation criteria involving integral averages of a(t) due to Wintner, Hartman, and recently Butler, Erbe and Mingarelli for the linear equation are shown to remain valid for the general equation, subject to certain nonlinear conditions on f(y). In particular, these results are therefore valid for the Emden-Fowler equation.

scholarly journals Generalized Weierstrass Integrability of a Class of Second-order Nonlinear Differential Equations

2021 ◽  
pp. 119-133
Author(s):  
Xin Zhao ◽  
Yanxia Hu
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Nonlinear Differential Equation ◽  
Traveling Wave ◽  
Nonlinear Differential Equations ◽  
Traveling Wave Solutions ◽  
Second Order ◽  
Wave Solutions ◽  
The Relationship ◽  
Integrating Factors

The generalized Weierstrass integrability of a class of second-order nonlinear differential equations is considered. The conditions of existence and the corresponding expressions of generalized Weierstrass inverse integrating factors of the second-order nonlinear differential equation are presented. The relationship between the generalized Weierstrass inverse integrating factors and the Weierstrass inverse integrating factors is given. Finally, as an application of the main results, a Kudryashov-Sinelshchikov equation for obtaining traveling wave solutions is considered.

scholarly journals Asymptotic Dichotomy in a Class of Third-Order Nonlinear Differential Equations with Impulses

2010 ◽  
Vol 2010 ◽  
pp. 1-20 ◽  
Author(s):  
Kun-Wen Wen ◽  
Gen-Qiang Wang ◽  
Sui Sun Cheng
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Nonlinear Differential Equation ◽  
Nonlinear Differential Equations ◽  
Functional Differential Equations ◽  
Higher Order ◽  
Third Order ◽  
Order Nonlinear Differential Equation ◽  
Functional Differential

Solutions of quite a few higher-order delay functional differential equations oscillate or converge to zero. In this paper, we obtain several such dichotomous criteria for a class of third-order nonlinear differential equation with impulses.

On Oscillation and Nonoscillation for Differential Equations with 𝑝-Laplacian

2007 ◽  
Vol 14 (2) ◽  
pp. 239-252
Author(s):  
Miroslav Bartušek ◽  
Mariella Cecchi ◽  
Zuzana Došlá ◽  
Mauro Marini
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Nonlinear Differential Equation ◽  
Oscillatory Solution ◽  
Second Order ◽  
Asymptotic Estimates ◽  
Nonoscillatory Solutions ◽  
Order Nonlinear Differential Equation ◽  
Monotonicity Properties ◽  
Oscillation And Nonoscillation

Abstract The existence of at least one oscillatory solution of a second order nonlinear differential equation with 𝑝-Laplacian is considered. The global monotonicity properties and asymptotic estimates for nonoscillatory solutions are investigated as well.

scholarly journals On the asymptotic behavior of solutions of certain third-order nonlinear differential equations

2005 ◽  
Vol 2005 (1) ◽  
pp. 29-35 ◽  
Author(s):  
Cemil Tunç
Keyword(s):  
Differential Equation ◽  
Asymptotic Behavior ◽  
Differential Equations ◽  
Nonlinear Differential Equation ◽  
Nonlinear Differential Equations ◽  
Sufficient Conditions ◽  
Asymptotic Behavior Of Solutions ◽  
Third Order ◽  
The Third ◽  
All Solutions

We establish sufficient conditions under which all solutions of the third-order nonlinear differential equation x ⃛+ψ(x,x˙,x¨)x¨+f(x,x˙)=p(t,x,x˙,x¨) are bounded and converge to zero as t→∞.

scholarly journals The Integrating Factors for Riccati and Abel Differential Equations

2015 ◽  
Vol 7 (2) ◽  
pp. 125
Author(s):  
Chein-Shan Liu
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Nonlinear Differential Equations ◽  
Type Systems ◽  
Second Order ◽  
Numerical Schemes ◽  
Riccati Differential Equation ◽  
Group Preserving Scheme ◽  
Abel Differential Equation ◽  
Abel Differential Equations

We can recast the Riccati and Abel differential equationsinto new forms in terms of introduced integrating factors.Therefore, the Lie-type systems endowing with transformation Lie-groups$SL(2,{\mathbb R})$ can be obtained.The solution of second-order linearhomogeneous differential equation is an integrating factorof the corresponding Riccati differential equation.The numerical schemes which are developed to fulfil the Lie-group property have better accuracy and stability than other schemes.We demonstrate that upon applying the group-preserving scheme (GPS) to the logistic differential equation, it is not only qualitatively correct for all values of time stepsize $h$, and is also the most accurate one among all numerical schemes compared in this paper.The group-preserving schemes derived for the Riccati differential equation, second-order linear homogeneous and non-homogeneous differential equations, the Abel differential equation and higher-order nonlinear differential equations all have accuracy better than $O(h^2)$.

scholarly journals On the correct formulation of a nonlinear differential equations in Banach space

2001 ◽  
Vol 26 (7) ◽  
pp. 437-444
Author(s):  
Mahmoud M. El-Borai ◽  
Osama L. Moustafa ◽  
Fayez H. Michael
Keyword(s):  
Differential Equation ◽  
Banach Space ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Nonlinear Differential Equation ◽  
Existence And Uniqueness ◽  
Initial Value Problems ◽  
Nonlinear Differential Equations ◽  
Initial Value ◽  
Correct Formulation

We study, the existence and uniqueness of the initial value problems in a Banach spaceEfor the abstract nonlinear differential equation(dn−1/dtn−1)(du/dt+Au)=B(t)u+f(t,W(t)), and consider the correct solution of this problem. We also give an application of the theory of partial differential equations.

scholarly journals Some of the methods used to solve complete and incomplete differential equations

2021 ◽  
Vol 12 (6) ◽  
pp. 193-200
Author(s):  
BASHEER ABD AL-RIDA SADIQ
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Second Order ◽  
First Order ◽  
Special Cases

This paper studies the methods used to solve complete and in complete differential equations and types of first order and second order and Exact differential equation to solve integration general in This equation Fur there more, and the Special cases to find the integration factor use solve those types of equations is use as well,supported by a relevant variety of examples.

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