Exact formulas for the accessory coefficients in the Schwarz equation

1984 ◽  
Vol 17 (3) ◽  
pp. 165-171 ◽  
Author(s):  
A. B. Venkov
Keyword(s):  
Schwarz Equation

scholarly journals On the problem of determining parameters in the Schwarz equation

2018 ◽  
Vol 25 (0) ◽  
pp. 50-62
Author(s):  
I. A. Kolesnikov
Keyword(s):  
Schwarz Equation

scholarly journals FURTHER GENERALISATIONS OF THE KUMMER-SCHWARZ EQUATION: ALGEBRAIC AND SINGULARITY PROPERTIES

2017 ◽  
Vol 57 (6) ◽  
pp. 467 ◽  
Author(s):  
R Sinuvasan ◽  
K Krishnakumar ◽  
K M Tamizhmani ◽  
PGL Leach
Keyword(s):  
Closed Form ◽  
New Class ◽  
Schwarz Equation

The Kummer–Schwarz Equation, 2<em>y'y'''</em>− 3(<em>y''</em>)<sup>2</sup> = 0, has a generalisation, (<em>n</em> − 1)<em>y</em><sup>(<em>n</em>−2)</sup><em>y</em><sup>(<em>n</em>)</sup> − <em>ny</em><sup>(<em>n</em>−1)<sup>2</sup></sup> = 0, which shares many properties with the parent form in terms of symmetry and singularity. All equations of the class are integrable in closed form. Here we introduce a new class, (<em>n</em>+q−2)<em>y</em><sup>(<em>n</em>−2</sup>)<em>y</em><sup>(<em>n</em>)</sup> −(<em>n</em>+<em>q</em>−1)<em>y</em><sup>(<em>n</em>−1)<sup>2</sup></sup> = 0, which has different integrability and singularity properties.

scholarly journals PROPERTIES OF A DIFFERENTIAL SEQUENCE BASED UPON THE KUMMER-SCHWARZ EQUATION

2020 ◽  
Vol 60 (5) ◽  
pp. 428-434
Author(s):  
Adhir Maharaj ◽  
Kostis Andriopoulos ◽  
Peter Leach
Keyword(s):  
Riccati Equation ◽  
Singularity Analysis ◽  
Recursion Operator ◽  
First Order ◽  
The Relationship ◽  
Schwarz Equation

In this paper, we determine a recursion operator for the Kummer-Schwarz equation, which leads to a sequence with unacceptable singularity properties. A different sequence is devised based upon the relationship between the Kummer-Schwarz equation and the first-order Riccati equation for which a particular generator has been found to give interesting and excellent properties. We examine the elements of this sequence in terms of the usual properties to be investigated – symmetries, singularity properties, integrability, alternate sequence – and provide an explanation of the curious relationship between the results of the singularity analysis and a consideration of the solution of each element obtained by quadratures.

On the Generalizations of the Kummer–Schwarz Equation

2020 ◽  
Vol 192 ◽  
pp. 111691 ◽  
Author(s):  
Yuri Dimitrov Bozhkov ◽  
Pammela Ramos da Conceição
Keyword(s):  
Schwarz Equation

Algebraic and Singularity Properties of a Class of Generalisations of the Kummer–Schwarz Equation

2016 ◽  
Vol 28 (2) ◽  
pp. 315-324
Author(s):  
R. Sinuvasan ◽  
K. M. Tamizhmani ◽  
P. G. L. Leach
Keyword(s):  
Schwarz Equation

scholarly journals Sistema óptimo, soluciones invariantes y clasificación completa del grupo de simetrías de Lie para la ecuación de Kummer-Schwarz generalizada y su representación del álgebra de Lie

2021 ◽  
Vol 39 (2) ◽  
Author(s):  
Danilo García Hernández ◽  
Oscar Mario Londoño Duque ◽  
Yeisson Acevedo ◽  
Gabriel Loaiza
Keyword(s):  
Symmetry Group ◽  
Lie Algebra ◽  
Complete Classification ◽  
Lie Symmetry ◽  
Invariant Solutions ◽  
Generating Operators ◽  
Alternative Solutions ◽  
Schwarz Equation

We obtain the complete classification of the Lie symmetry group and the optimal system’s generating operators associated with a particular case of the generalized Kummer - Schwarz equation. Using those operators we characterize all invariant solutions, alternative solutions were found for the equation studied and the Lie algebra associated with the symmetry group is classified.

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