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

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Abstract

The Kummer–Schwarz Equation, \(2 y'y''' - 3 y''{}^2 = 0\), (the prime denotes differentiation with respect to the independent variable x) is well known from its connection to the Schwartzian Derivative and in its own right for its interesting properties in terms of symmetry and singularity. We examine a class of equations which are a natural generalisation of the Kummer–Schwarz Equation and find that the algebraic and singularity properties of this class of equations display an attractive set of patterns. We demonstrate that all members of this class are readily integrable.

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Notes

  1. For the calculation of the symmetries we use the Mathematica add-on Sym [3–6].

  2. For a linear equation of order n the three elements are usually written as \(\partial _x\), \(x\partial _x+\frac{(n-1)}{2}{} y\partial _y\) and \(x^2\partial _x + (n-1) xy\partial _y\).

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Acknowledgments

R. Sinuvasan thanks the University Grants Commission [grant no. F. 17-115/98 (SA-1)] for its support. PGLL thanks Professor KM Tamizhmani and the Department of Mathematics, University of Pondicherry, for the provision of facilities whilst this work was undertaken. PGLL also thanks the University of KwaZulu-Natal and the National Research Foundation of the Republic of South Africa for their continued support. Any views expressed in this paper are not necessarily those of the two institutions.

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Authors and Affiliations

  1. Department of Mathematics, Pondicherry University, Kalapet, 605 014, India

    R. Sinuvasan & K. M. Tamizhmani

  2. School of Mathematics, Statistics and Computer Science, University of KwaZulu-Natal, Private Bag X54001, Durban, 4000, Republic of South Africa

    P. G. L. Leach

  3. Department of Mathematics, Institute for Systems Science, Durban University of Technology, POB 1334, Durban, 4000, Republic of South Africa

    P. G. L. Leach

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  1. R. Sinuvasan
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  2. K. M. Tamizhmani
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  3. P. G. L. Leach
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Correspondence to R. Sinuvasan.

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Sinuvasan, R., Tamizhmani, K.M. & Leach, P.G.L. Algebraic and Singularity Properties of a Class of Generalisations of the Kummer–Schwarz Equation. Differ Equ Dyn Syst 28, 315–324 (2020). https://doi.org/10.1007/s12591-016-0327-5

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