A New Approach to Solving Periodic Differential Systems

2021 ◽  
Vol 1 ◽  
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Second Order ◽  
Oscillatory Motion ◽  
Differential Systems ◽  
New Approach

Mathematicians and physicists are well acquainted with second-order ordinary differential equations (ODE), the most prominent of them being the class of equations that govern oscillatory motion.

Oscillation of second order half-linear neutral differential equations with weaker restrictions on shifted arguments

2020 ◽  
Vol 70 (2) ◽  
pp. 389-400
Author(s):  
Simona Fišnarová ◽  
Robert Mařík
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Second Order ◽  
New Approach ◽  
Oscillation Criteria ◽  
Neutral Differential Equations ◽  
Oscillatory Properties

Abstract Neutral differential equations are one of the most important extensions of classical ordinary differential equations and aim to give a better explanation for modeling phenomena where ordinary differential equations are insufficient. Naturally, all the questions studied in the scope of ordinary differential equations attracted the attention also for neutral differential equations. In this paper we study the oscillatory properties of second order half-linear neutral differential equations. We present oscillation criteria derived using a new approach. This approach allows us to reduce common restrictions on the deviations in arguments which are present in the currently known results of this type.

scholarly journals Second-order impulsive differential systems with mixed and several delays

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Shyam Sundar Santra ◽  
Apurba Ghosh ◽  
Omar Bazighifan ◽  
Khaled Mohamed Khedher ◽  
Taher A. Nofal
Keyword(s):  
Differential Equations ◽  
Sufficient Conditions ◽  
Impulsive Differential Equations ◽  
Oscillation Theory ◽  
Second Order ◽  
Neutral Delay ◽  
Differential Systems ◽  
Impulsive Differential Systems ◽  
Several Delays ◽  
Necessary And Sufficient

AbstractIn this work, we present new necessary and sufficient conditions for the oscillation of a class of second-order neutral delay impulsive differential equations. Our oscillation results complement, simplify and improve recent results on oscillation theory of this type of nonlinear neutral impulsive differential equations that appear in the literature. An example is provided to illustrate the value of the main results.

The Geometry of d2y1/dt2 = f (y, ẏ, t) and d2y2/dt2 = g(y, ẏ, t), and Euclidean Spaces

2006 ◽  
Vol 49 (2) ◽  
pp. 170-184
Author(s):  
Richard Atkins
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Shortest Paths ◽  
Equivalence Problem ◽  
Second Order ◽  
System Of Differential Equations ◽  
Lagrange Equations ◽  
Invariant Functions ◽  
Underlying Geometry ◽  
The Relationship

AbstractThis paper investigates the relationship between a system of differential equations and the underlying geometry associated with it. The geometry of a surface determines shortest paths, or geodesics connecting nearby points, which are defined as the solutions to a pair of second-order differential equations: the Euler–Lagrange equations of the metric. We ask when the converse holds, that is, when solutions to a system of differential equations reveals an underlying geometry. Specifically, when may the solutions to a given pair of second order ordinary differential equations d2y1/dt2 = f (y, ẏ, t) and d2y2/dt2 = g(y, ẏ, t) be reparameterized by t → T(y, t) so as to give locally the geodesics of a Euclidean space? Our approach is based upon Cartan's method of equivalence. In the second part of the paper, the equivalence problem is solved for a generic pair of second order ordinary differential equations of the above form revealing the existence of 24 invariant functions.

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