Aequationes Mathematicae 17 (1978) 148-153 Birkh~iuser Verlag, Basel University of Waterloo On solving certain differential equations with variable coefficients MOURAD E. H. ISMAIL Abs~racr We show how to solve certain types of linear ordinary differential equations with variable coefficients by using Appell polynomials. 1. Introduction Poli [4] showed that yH, (x) = t k 0t,_k, k=0 (1.1) where y is the generating function for the Hermite polynomials {H.(x)}~=o, that is tn eXt_t2/2" r = y(x, t) = g.(x) ~. = n=0 The Hermite polynomials are orthogonal on the real line, hence have real and simple zeros. This observation shows that a fundamental system of solutions to t tkD'~-~y = O, D~ =- dt' k =O n o is {y(xj, )}j=l where xl, x2,.. x, are the zeros of H,(x). Haradze [3] derived an analogue of (1.1) for the ultraspherical polynomials and later Allaway [2] estab- lished the analogues results for the Laguerre, Meixner and Charlier polynomials. All these polynomials are orthogonal. We do not think that orthogonality has anything to do with the existence of AIdS (1970) subject classification: Primary 33A70, 34A05. Secondary 39A15. Received June 7, 1975 and, in revised form, July 23, I976.