The equation for a product of solutions of two second-order linear ODEs

Abstract

The following problem is studied. Consider two linear homogeneous second-order ordinary differential equations of the form ry''+r'y'=fy (eqn.1) and ru''+r'u'=gu (eqn.2). These equations are chosen to be formally self-adjoint. The function /spl upsi/(z) is defined as a product of the arbitrary solutions y(z) and g(z) of these equations. /spl upsi/:=yu. It is assumed that the functions r(z), f(z), and g(z) are analytical functions. Moreover, if applications to special functions are studied then r(z) may be taken a polynomial, and f(z) g(z) are fractions of two polynomials. The question arises: what is the differential equation for which the function /spl upsi/(z) is a solution? A more sophisticated question is: is there a differential equation for which singularities are located only at the points where singularities of eqs. 1 and 2 are? These are discussed.