scholarly journals Integral averaging techniques for the oscillation and nonoscillation of solutions of second order ordinary differential equations

1994 ◽  
Vol 24 (3) ◽  
pp. 657-670 ◽  
Author(s):  
Manabu Naito
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Second Order ◽  
Oscillation And Nonoscillation ◽  
Averaging Techniques

scholarly journals Integral averaging techniques for the oscillation of second order sublinear ordinarry differential equations

1986 ◽  
Vol 40 (1) ◽  
pp. 111-130 ◽  
Author(s):  
Ch. G. Philos
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Second Order ◽  
Averaging Technique ◽  
Oscillation Criteria ◽  
Integral Averaging Technique ◽  
Special Cases ◽  
Averaging Techniques

AbstractNew oscillation criteria are established for second order sublinear ordinary differential equations with alternating coefficients. These criteria are obtained by using an integral averaging technique and can be applied in some special cases in which other classical oscillation results are no applicable.

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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