Integrals of the Differential Equations ẍ + f ( s ) x = 0

1967 ◽  
Vol 10 (2) ◽  
pp. 191-196 ◽  
Author(s):  
R. Datko
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Stability Problem ◽  
Additional Assumption ◽  
Order Differential Equation ◽  
Second Order ◽  
Extensive Survey ◽  
Second Order Differential Equation ◽  
All Solutions ◽  
New Criterion

In this note we consider a relatively ancient stability problem: the behaviour of solutions of the second order differential equation ẍ + f(s) x = 0, where f(s) tends to plus infinity as s tends to plus infinity. An extensive survey of the literature concerning this problem and a resume of results may be found in [ l ]. More recently McShane et a l. [2] have shown that the additional assumption f(s) ≥ 0 is not sufficient to guarantee that all solutions tend to zero as s tends to infinity. Our aim is to demonstrate a new criterion for which all solutions do have the above property. This criterion overlaps many of the cases heretofore considered.

On the integrability of solutions of perturbed non-linear differential equations

1977 ◽  
Vol 77 (3-4) ◽  
pp. 309-318 ◽  
Author(s):  
Paul W. Spikes
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Order Differential Equation ◽  
Sufficient Conditions ◽  
Second Order ◽  
Linear Differential Equations ◽  
Second Order Differential Equation ◽  
All Solutions ◽  
Non Linear ◽  
Linear Differential

SynopsisSufficient conditions are given to insure that all solutions of a perturbed non-linear second-order differential equation have certain integrability properties. In addition, some continuability and boundedness results are given for solutions of this equation.

scholarly journals Some new results on the stability and boundedness of solutions of certain class of second order vector differential equations

2021 ◽  
Vol 7 (1) ◽  
pp. 108-115
Author(s):  
A.A. ADEYANJU ◽  
◽  
D.O. ADAMS ◽  
Keyword(s):  
Differential Equation ◽  
Lyapunov Function ◽  
Differential Equations ◽  
Order Differential Equation ◽  
Zero Solution ◽  
Second Order ◽  
Boundedness Of Solutions ◽  
Second Order Differential Equation ◽  
All Solutions ◽  
The Stability

n this paper, we provide certain conditions that guarantee the stability of the zero solution when P(t, X, Y) = 0 and boundedness of all solutions when P(t, X, Y)# 0 of a certain system of second order differential equation using a suitable Lyapunov function. The results in this paper are quite new and complement those in the literature. Examples are given to demonstrate the correctness of the established results.

A mechanical method for the solution of second-order linear differential equations

1931 ◽  
Vol 27 (4) ◽  
pp. 546-552 ◽  
Author(s):  
E. C. Bullard ◽  
P. B. Moon
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Boundary Conditions ◽  
Order Differential Equation ◽  
Second Order ◽  
Linear Differential Equations ◽  
Mechanical Method ◽  
Order Linear ◽  
Second Order Differential Equation ◽  
Linear Differential

A mechanical method of integrating a second-order differential equation, with any boundary conditions, is described and its applications are discussed.

scholarly journals Periodic solutions for periodic second-order differential equations with variable potentials

2018 ◽  
Vol 24 (2) ◽  
pp. 127-137
Author(s):  
Jaume Llibre ◽  
Ammar Makhlouf
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Periodic Solutions ◽  
Order Differential Equation ◽  
Sufficient Conditions ◽  
Second Order ◽  
Second Order Differential Equations ◽  
Second Order Differential Equation ◽  
Existence Of Periodic Solutions

Abstract We provide sufficient conditions for the existence of periodic solutions of the second-order differential equation with variable potentials {-(px^{\prime})^{\prime}(t)-r(t)p(t)x^{\prime}(t)+q(t)x(t)=f(t,x(t))} , where the functions {p(t)>0} , {q(t)} , {r(t)} and {f(t,x)} are {\mathcal{C}^{2}} and T-periodic in the variable t.

scholarly journals Solutions of the perturbed KdV equation for convecting fluids by factorizations

Open Physics ◽  
2010 ◽  
Vol 8 (4) ◽  
Author(s):  
Octavio Cornejo-Pérez ◽  
Haret Rosu
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Order Differential Equation ◽  
Kdv Equation ◽  
Travelling Wave ◽  
Second Order ◽  
Travelling Wave Solutions ◽  
Second Order Differential Equation ◽  
Wave Solutions ◽  
Perturbed Kdv Equation

AbstractIn this paper, we obtain some new explicit travelling wave solutions of the perturbed KdV equation through recent factorization techniques that can be performed when the coefficients of the equation fulfill a certain condition. The solutions are obtained by using a two-step factorization procedure through which the perturbed KdV equation is reduced to a nonlinear second order differential equation, and to some Bernoulli and Abel type differential equations whose solutions are expressed in terms of the exponential andWeierstrass functions.

scholarly journals Weierstrass elliptic difference equations

1987 ◽  
Vol 35 (1) ◽  
pp. 43-48 ◽  
Author(s):  
Renfrey B. Potts
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Difference Equations ◽  
Elliptic Function ◽  
Order Differential Equation ◽  
Second Order ◽  
Elliptic Difference ◽  
First Order ◽  
Second Order Differential Equation ◽  
Weierstrass Elliptic Function

The Weierstrass elliptic function satisfies a nonlinear first order and a nonlinear second order differential equation. It is shown that these differential equations can be discretized in such a way that the solutions of the resulting difference equations exactly coincide with the corresponding values of the elliptic function.

scholarly journals New Existence of Fixed Point Results in Generalized Pseudodistance Functions with Its Application to Differential Equations

Mathematics ◽  
2018 ◽  
Vol 6 (12) ◽  
pp. 324 ◽  
Author(s):  
Sujitra Sanhan ◽  
Winate Sanhan ◽  
Chirasak Mongkolkeha
Keyword(s):  
Differential Equation ◽  
Fixed Point ◽  
Differential Equations ◽  
Contraction Mapping ◽  
Fixed Point Theorems ◽  
Metric Spaces ◽  
Order Differential Equation ◽  
Second Order ◽  
Second Order Differential Equation ◽  
Generalized Pseudodistance

The purpose of this article is to prove some existences of fixed point theorems for generalized F -contraction mapping in metric spaces by using the concept of generalized pseudodistance. In addition, we give some examples to illustrate our main results. As the application, the existence of the solution of the second order differential equation is given.

scholarly journals Some properties of orthogonal polynomials satisfying fourth order differential equations

1995 ◽  
Vol 37 (1) ◽  
pp. 105-113 ◽  
Author(s):  
R. G. Campos ◽  
L. A. Avila
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Orthogonal Polynomials ◽  
Order Differential Equation ◽  
Polynomial Solution ◽  
Second Order ◽  
System Of Nonlinear Equations ◽  
Singular Boundary ◽  
Classical Orthogonal Polynomials ◽  
Second Order Differential Equation

In the last few years, there has been considerable interest in the properties of orthogonal polynomials satisfying differential equations (DE) of order greater than two, their connection to singular boundary value problems, their generalizations, and their classification as solutions of second order DE (see for instance [2–8]). In this last interesting problem, some known facts about the classical orthogonal polynomials can be incorporated to connect these two sets of families and yield some nontrivial results in a very simple way. In this paper we only work with the nonclassical Jacobi type, Laguerre type and Legendre type polynomials, and we show how they can be connected with the classical Jacobi, Laguerre and Legendre polynomials, respectively; at the same time we obtain certain bounds for the zeros of the first ones by using a system of nonlinear equations satisfied by the zeros of any polynomial solution of a second order differential equation which, for the classical polynomials is known since Stieltjes and concerns the electrostatic interpretation of the zeros [10, Section 6.7; 9,1]. We also correct an expression for the second order differential equation of the Legendre type polynomials that circulates through the literature.

Note on the Uniqueness of the Green'S Functions Associated With Certain Differential Equations

1950 ◽  
Vol 2 ◽  
pp. 314-325 ◽  
Author(s):  
D. B. Sears
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Green’S Function ◽  
Green's Function ◽  
Green's Functions ◽  
Order Differential Equation ◽  
Green’S Functions ◽  
Second Order ◽  
Second Order Differential Equation

Conditions to be imposed on q(x) which ensure the uniqueness of the Green's function associated with the linear second-order differential equation

Nonprincipal solutions in oscillation criteria for half-linear differential equations

2010 ◽  
Vol 47 (1) ◽  
pp. 127-137
Author(s):  
Ondřej Došlý ◽  
Jana Řezníčková
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Order Differential Equation ◽  
Second Order ◽  
Oscillation Criterion ◽  
Linear Differential Equations ◽  
Integral Term ◽  
Oscillation Criteria ◽  
Second Order Differential Equation ◽  
Linear Differential

We establish a new oscillation criterion for the half-linear second order differential equation \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$(r(t)\Phi (x'))' + c(t)\Phi (x) = 0,\Phi (x): = |x|^{p - 2} x,p > 1.$$ \end{document} In this criterion, an integral term appears which involves a nonprincipal solution of a certain equation associated with (*).

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