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

Abstract

The Kummer–Schwarz Equation, 2y'y'''− 3(y'')² = 0, has a generalisation, (n − 1)y⁽ⁿ⁻²⁾y⁽ⁿ⁾ − ny⁽ⁿ⁻¹⁾² = 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, (n+q−2)y⁽ⁿ⁻²)y⁽ⁿ⁾ −(n+q−1)y⁽ⁿ⁻¹⁾² = 0, which has different integrability and singularity properties.