Cheng Equation: A Revisit Through Symmetry Analysis

The symmetry analysis of the Cheng Equation is performed. The Cheng Equation is reduced to a first-order equation of either Abel's Equations, the analytic solution of which is given in terms of special functions. Moreover, for a particular symmetry the system is reduced to the Riccati Equation or to the linear nonhomogeneous equation of Euler type. Henceforth, the general solution of the Cheng Equation with the use of the Lie theory is discussed, as also the application of Lie symmetries in a generalized Cheng equation.Comment: 10 pages. Accepted for publication in Quaestiones Mathematicae journa

Some properties of the Nesterenko differential sequence

We present some properties of a differential sequence prompted by a formula in a paper by Maryna Nesterenko (International Journal of Mathematics and Mathematical Sciences 2006 (2006), Article ID 17410) in terms of the symmetry, singularity and integrability properties of the elements of the sequence.Mathematics Subject Classification (2010): 34M35, 34M45, 34M55.Keywords: Differential sequence, symmetry analysis, singularity analysis, integrabilit

Solution of the Master Equation for Quantum Brownian Motion Given by the Schrödinger Equation

We consider the master equation of quantum Brownian motion, and with the application of the group invariant transformation, we show that there exists a surface on which the solution of the master equation is given by an autonomous one-dimensional Schrödinger Equation

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

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