scholarly journals A converse of Sturm's separation theorem

2021 ◽  
pp. 1-8
Author(s):  
Leila Gholizadeh ◽  
Angelo Mingarelli
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Principal Part ◽  
Turning Point ◽  
Second Order ◽  
Separation Theorem ◽  
Linearly Independent

We show that Sturm's classical separation theorem on the interlacing of the zeros of linearly independent solutions of real second order two-term ordinary differential equations necessarily fails in the presence of a turning point in the principal part of the equation. Related results are discussed.

scholarly journals The Sturm Separation Theorem for Impulsive Delay Differential Equations

2018 ◽  
Vol 71 (1) ◽  
pp. 65-70
Author(s):  
Alexander Domoshnitsky ◽  
Vladimir Raichik
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Delay Differential Equations ◽  
Independent Solution ◽  
Delay Equations ◽  
Separation Theorem ◽  
Linearly Independent ◽  
Impulsive Delay Differential Equations ◽  
Impulsive Delay ◽  
Delay Differential

Abstract Wronskian is one of the classical objects in the theory of ordinary differential equations. Properties of Wronskian lead to important conclusions on behaviour of solutions of delay equations. For instance, non-vanishing Wronskian ensures validity of the Sturm separation theorem (between two adjacent zeros of any solution there is one and only one zero of every other nontrivial linearly independent solution) for delay equations. We propose the Sturm separation theorem in the case of impulsive delay differential equations and obtain assertions about its validity.

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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