scholarly journals Boundedness of Solutions to Differential Equations of Fourth Order with Oscillatory Restoring and Forcing Terms

2013 ◽  
Vol 2013 ◽  
pp. 1-5 ◽  
Author(s):  
Cemil Tunç ◽  
Muzaffer Ateş
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Nonlinear Differential Equation ◽  
Fourth Order ◽  
Cauchy Formula ◽  
Boundedness Of Solutions ◽  
Constant Coefficients ◽  
Particular Solution

This paper deals with the boundedness of solutions to a nonlinear differential equation of fourth order. Using the Cauchy formula for the particular solution of nonhomogeneous differential equations with constant coefficients, we prove that the solution and its derivatives up to order three are bounded.

scholarly journals On the Oscillatory Behavior of a Class of Fourth-Order Nonlinear Differential Equation

Symmetry ◽  
2020 ◽  
Vol 12 (4) ◽  
pp. 524 ◽  
Author(s):  
Osama Moaaz ◽  
Poom Kumam ◽  
Omar Bazighifan
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Nonlinear Differential Equation ◽  
Fourth Order ◽  
Oscillatory Behavior ◽  
Oscillation Criteria ◽  
Order Nonlinear Differential Equation

In this work, we study the oscillatory behavior of a class of fourth-order differential equations. New oscillation criteria were obtained by employing a refinement of the Riccati transformations. The new theorems complement and improve a number of results reported in the literature. An example is provided to illustrate the main results.

scholarly journals Determining a Particular Solution for n-th Order Linear Differential Equations with Constant Coefficients

2017 ◽  
Vol 23 (3) ◽  
pp. 24-29
Author(s):  
Vasile Căruțașu
Keyword(s):  
Differential Equation ◽  
General Solution ◽  
Differential Equations ◽  
Linear Differential Equation ◽  
Linear Differential Equations ◽  
Free Term ◽  
Order Linear ◽  
Constant Coefficients ◽  
Particular Solution ◽  
Linear Differential

Abstract For n-th order linear differential equations with constant coefficients, the problem to be solved is related to determining a particular solution, and then, with the general solution of n-th homogeneous linear differential equation with constant coefficients attached, to write the general solution of n-th linear differential equation with the given constant coefficients. In all the works that deal with this issue three situations are analyzed: the situation in which the free term is a polynomial P(x), the situation in which the free term is like P(x)· eα·x and lastly, the situation in which the free term is like eω·x · (P(x)· cos(β·x)+ Q(x)·sin(β·x)). In this study we aim to analyze if the free term is a combination of the three cases mentioned.

scholarly journals SOLUSI DARI PERSAMAAN CAUCHY–EULER NONHOMOGEN KASUS LOGARITMIK

2020 ◽  
Vol 9 (2) ◽  
pp. 125
Author(s):  
I GEDE PUTU MIKI SUKADANA ◽  
I NYOMAN WIDANA ◽  
KETUT JAYANEGARA
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Parameter Variation ◽  
Undetermined Coefficient ◽  
Constant Coefficients ◽  
Particular Solution ◽  
Ordinary Solution ◽  
The Right ◽  
Coefficient Method ◽  
Undetermined Coefficient Method

Ordinary differential equation is one form of differential equations that are often found in everyday life. One form of ordinary differential equations which has non–constant coefficients is the Cauchy–Euler differential equation. In the nonhomogeneous Cauchy–Euler differential equations, the undetermined coefficient and the parameter variation were the most method that often used to find the particular solution. This paper aimed to show a new solution that was shorter than the previous methods for nonhomogeneous Cauchy–Euler differential equations with the right side was a logarithmic form. The new solution had been proven to produce the same solution as the ordinary solution sought using the undetermined coefficient method.

Nonlinear Oscillation of Fourth Order Differential Equations

1976 ◽  
Vol 28 (4) ◽  
pp. 840-852 ◽  
Author(s):  
Takaŝi Kusano ◽  
Manabu Naito
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Nonlinear Differential Equation ◽  
Fourth Order ◽  
Nonlinear Oscillation ◽  
Order Nonlinear Differential Equation ◽  
Image Position

In this paper we are concerned with the fourth order nonlinear differential equationwhere the following conditions are always assumed to hold:(a) r(t) is continuous and positive for t ≠ 0, and

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.

Self-Excited Oscillations in Dynamical Systems Possessing Retarded Actions

1942 ◽  
Vol 9 (2) ◽  
pp. A65-A71 ◽  
Author(s):  
Nicholas Minorsky
Keyword(s):  
Differential Equation ◽  
Dynamical Systems ◽  
Differential Equations ◽  
Linear Equation ◽  
Finite Order ◽  
Characteristic Equation ◽  
High Derivative ◽  
Time Lags ◽  
Constant Coefficients ◽  
The Subject

Abstract There exists a variety of dynamical systems, possessing retarded actions, which are not entirely describable in terms of differential equations of a finite order. The differential equations of such systems are sometimes designated as hysterodifferential equations. An important particular case of such equations, encountered in practice, is when the original differential equation for unretarded quantities is a linear equation with constant coefficients and the time lags are constant. The characteristic equation, corresponding to the hysterodifferential equation for retarded quantities in such a case, has a series of subsequent high-derivative terms which generally converge. It is possible to develop a simple graphical interpretation for this equation. Such systems with retarded actions are capable of self-excitation. Self-excited oscillations of this character are generally undesirable in practice and it is to this phase of the subject that the present paper is devoted.

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→∞.

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.

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