  We analyze the polynomial solutions of the linear differential equation
$p_2(x)y''+p_1(x)y'+p_0(x)y=0$ where $p_j(x)$ is a $j^{\rm th}$-degree
polynomial. We discuss all the possible polynomial solutions and their
dependence on the parameters of the polynomials $p_j(x)$. Special cases are
related to known differential equations of mathematical physics. Classes of new
soluble problems are exhibited. General results are obtained for weight
functions and orthogonality relations.
