On Euler-Imshenetsky-Darboux transformation of second-order linear differential equations

Abstract

It is shown, how to generate infinite sequences of differential equations of the second order based on some standard equations, using Euler-Imshenetsky-Darboux (EID) transformation. For all this, factorizations of differential operators and operational identities are used.Some generalizations of integrable cases of Schrödinger's equations are finded. The example of integrable equation with liouvillian coefficients, that, apparently, can not be solved by Singer and Kovacic's algorithm (and its modifications), was built. The alogorithm and the program for solving conctructed classes of equations, was realized in REDUCE system. Corresponding procedure GENERATE is addendum to the ODESOLVE procedure of REDUCE system. The results of using GENERATE procedure (REDUCE 3.8) was compared with results of DSolve procedure (Maple 10). Though, algorithm, based on EID transformation is not an alternative to the unviersal ones, in the borders of its applicability, its very powerful.