 
Summary: Polynomial Solutions of Differential Equations
H. Azad, A. Laradji and M. T. Mustafa
Department of Mathematics and Statistics
King Fahd University of Petroleum & Minerals
Dhahran, Saudi Arabia
hassanaz@kfupm.edu.sa, alaradji@kfupm.edu.sa, tmustafa@kfupm.edu.sa
Abstract
A new approach for investigating polynomial solutions of differential equa
tions is proposed. It is based on elementary linear algebra. Any differential
operator of the form L(y) =
k=N
k=0
ak(x)y(k)
, where ak is a polynomial of degree
k, over an infinite ground field F has all eigenvalues in F in the space of
polynomials of degree at most n, for all n. If these eigenvalues are distinct, then
there is a unique monic polynomial of degree n which is an eigenfunction of the
operator L for every nonnegative integer n.
Specializing to the real field, the potential of the method is illustrated by
recovering Bochner's classification of second order ODEs with polynomial coef
