The exact difference equation of the first order

1933 ◽  
Vol 29 (3) ◽  
pp. 373-381
Author(s):  
L. M. Milne-Thomson
Keyword(s):  
Differential Equation ◽  
Difference Equation ◽  
Linear Equation ◽  
Sufficient Conditions ◽  
First Order ◽  
Non Linear Equation ◽  
Exact Difference ◽  
The Difference ◽  
Necessary And Sufficient ◽  
Limiting Case

The theory of the exact difference equation in the general linear case has been fully developed, but the corresponding theory for the non-linear equation of the first order does not appear to have been considered. In this paper necessary and sufficient conditions for the difference equation of the first order to be exact and the form of the primitive are obtained. It appears that two conditions are required for a difference equation to be exact, one of which is identically satisfied in the limiting case of the exact differential equation. These conditions are applied to determining the primitive in some cases where the conditions for exactness are not satisfied.

scholarly journals Existence of a Period-Two Solution in Linearizable Difference Equations

2013 ◽  
Vol 2013 ◽  
pp. 1-9
Author(s):  
E. J. Janowski ◽  
M. R. S. Kulenović
Keyword(s):  
Difference Equation ◽  
Linear Equation ◽  
Difference Equations ◽  
Initial Conditions ◽  
Sufficient Conditions ◽  
Minimal Period ◽  
Real Numbers ◽  
Existence And Nonexistence ◽  
The Difference ◽  
Necessary And Sufficient

Consider the difference equationxn+1=f(xn,…,xn−k),n=0,1,…,wherek∈{1,2,…}and the initial conditions are real numbers. We investigate the existence and nonexistence of the minimal period-two solution of this equation when it can be rewritten as the nonautonomous linear equationxn+l=∑i=1−lkgixn−i,n=0,1,…,wherel,k∈{1,2,…}and the functionsgi:ℝk+l→ℝ. We give some necessary and sufficient conditions for the equation to have a minimal period-two solution whenl=1.

scholarly journals Necessary and sufficient conditions for oscillation of first order neutral delay difference equations

2015 ◽  
Vol 3 (2) ◽  
pp. 61
Author(s):  
A. Murgesan ◽  
P. Sowmiya
Keyword(s):  
Difference Equation ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Delay Difference Equation ◽  
Neutral Delay ◽  
First Order ◽  
Constant Coefficients ◽  
Necessary And Sufficient ◽  
Delay Difference Equations ◽  
Delay Difference

<p>In this paper, we obtained some necessary and sufficient conditions for oscillation of all the solutions of the first order neutral delay difference equation with constant coefficients of the form <br />\begin{equation*} \quad \quad \quad \quad \Delta[x(n)-px(n-\tau)]+qx(n-\sigma)=0, \quad \quad n\geq n_0 \quad \quad \quad \quad \quad \quad {(*)} \end{equation*}<br />by constructing several suitable auxiliary functions. Some examples are also given to illustrate our results.</p>

scholarly journals Oscillation and nonoscillation of a delay differential equation

1994 ◽  
Vol 49 (1) ◽  
pp. 69-79 ◽  
Author(s):  
Chunhai Kou ◽  
Weiping Yan ◽  
Jurang Yan
Keyword(s):  
Differential Equation ◽  
Delay Differential Equation ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
First Order ◽  
Oscillating Coefficients ◽  
Image Position ◽  
Delay Differential ◽  
Necessary And Sufficient ◽  
Oscillation And Nonoscillation

In this paper, some necessary and sufficient conditions for oscillation of a first order delay differential equation with oscillating coefficients of the formare established. Several applications of our results improve and generalise some of the known results in the literature.

scholarly journals Property of oscillation of first order impulsive neutral differential equations with positive and negative coefficients

2019 ◽  
Vol 11 (1) ◽  
Author(s):  
Hussain Ali Mohamad ◽  
Aqeel Falih Jaddoa
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Sufficient Conditions ◽  
Variable Coefficients ◽  
Neutral Differential Equations ◽  
First Order ◽  
Positive And Negative Coefficients ◽  
Variable Delays ◽  
Impulsive Neutral Differential Equations ◽  
Necessary And Sufficient

            In this paper, necessary and sufficient conditions for oscillation are obtained, so that every solution of the linear impulsive neutral differential equation with variable delays and variable coefficients oscillates. Most of authors who study the oscillatory criteria of impulsive neutral differential equations, investigate the case of constant delays and variable coefficients. However the points of impulsive in this paper are more general. An illustrate example is given to demonstrate our claim and explain the results.

4. On the Linear Differential Equation of the Second Order

1878 ◽  
Vol 9 ◽  
pp. 93-98 ◽  
Author(s):  
Tait
Keyword(s):  
Differential Equation ◽  
Linear Differential Equation ◽  
Linear Equation ◽  
Arbitrary Constant ◽  
Second Order ◽  
General Linear ◽  
First Order ◽  
Non Linear Equation ◽  
Non Linear ◽  
Linear Differential

This paper contains the substance of investigations made for the most part many years ago, but recalled to me during last summer by a question started by Sir W. Thomson, connected with Laplace's theory of the tides.A comparison is instituted between the results of various processes employed to reduce the general linear differential equation of the second order to a non-linear equation of the first order. The relation between these equations seems to be most easily shown by the following obvious process, which I lit upon while seeking to integrate the reduced equation by finding how the arbitrary constant ought to be involved in its integral.

Design and Analysis of an oil Cushion

1973 ◽  
Vol 15 (1) ◽  
pp. 48-52 ◽  
Author(s):  
M. A. Satter
Keyword(s):  
Differential Equation ◽  
Linear Differential Equation ◽  
Linear Equation ◽  
Dynamic Characteristics ◽  
Order Linear Differential Equation ◽  
First Order ◽  
Non Linear Equation ◽  
Non Linear ◽  
Linear Differential ◽  
Good Agreement

The dynamic characteristics of an oil cushion, which was originally designed to eliminate impactive excitation to a mechanical lever and thereby achieve noise reduction, have been studied both theoretically and experimentally. The system motion is represented by a second order non-linear differential equation which can be reduced to a first order linear differential equation by changing the variables. An approximate but simple solution to the non-linear equation has also been presented. Theoretical and experimental results have good agreement.

scholarly journals Linearization of Fifth-Order Ordinary Differential Equations by Generalized Sundman Transformations

2018 ◽  
Vol 2018 ◽  
pp. 1-17
Author(s):  
Supaporn Suksern ◽  
Kwanpaka Naboonmee
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Differential Equations ◽  
Linear Equation ◽  
Ordinary Differential Equations ◽  
Sufficient Conditions ◽  
Order Ordinary Differential Equation ◽  
Linearization Problem ◽  
Necessary And Sufficient ◽  
Fifth Order

In this article, the linearization problem of fifth-order ordinary differential equation is presented by using the generalized Sundman transformation. The necessary and sufficient conditions which allow the nonlinear fifth-order ordinary differential equation to be transformed to the simplest linear equation are found. There is only one case in the part of sufficient conditions which is surprisingly less than the number of cases in the same part for order 2, 3, and 4. Moreover, the derivations of the explicit forms for the linearizing transformation are exhibited. Examples for the main results are included.

scholarly journals A note on a paper of Bowcock and Yu

1989 ◽  
Vol 40 (3) ◽  
pp. 421-424
Author(s):  
I. P. Stavroulakis
Keyword(s):  
Differential Equation ◽  
Order Differential Equation ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
First Order ◽  
All Solutions ◽  
Necessary And Sufficient

Consider the first order differential equation (1) , where pi, and τi, for i = 1,…,n, are positive constants. To find necessary and sufficient conditions, in terms of the coefficients and the delays only, under which all solutions of (1) oscillate, is a problem of great importence. In a recent paper, Bowcock and Yu claimed that is a necessary and sufficient condition for all solutions of (1) to be oscillatory. In this paper a counterexample shows that the above result is not valid and the error in this paper is indicated.

A Liapunov functional for a scalar differential difference equation

1978 ◽  
Vol 79 (3-4) ◽  
pp. 307-316 ◽  
Author(s):  
E. F. Infante ◽  
J. A. Walker
Keyword(s):  
Asymptotic Stability ◽  
Difference Equation ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Quadratic Functional ◽  
Liapunov Functional ◽  
Differential Difference Equation ◽  
The Difference ◽  
Rates Of Decay ◽  
Necessary And Sufficient

SynopsisGiven the scalar, retarded differential difference equation x'(t)=ax(t) +bx(t−τ), a quadratic functional in explicit form is obtained that yields necessary and sufficient conditions for the asymptotic stability of this equation. This functional a Liapunov functional, is obtained through the study of the Liapunov functions associated with a difference equation approximation of the difference differential equation. The functional then obtained not only yields necessary and sufficient conditions for asymptotic stability, but provides estimates for rates of decay of the solutions as well as conditions, for asymptotic stability independent of the magnitude of the delay τ.

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