Electronic Journal of Differential Equations,
Vol. 2002(2002), No. 12, pp. 1-14.
Title: Analytic solutions of n-th order differential equations at a
singular point
Author: Brian Haile (Northwest Missouri State Univ., Maryville, MO, USA)
Abstract:
Necessary and sufficient conditions are be given for the
existence of analytic solutions of the nonhomogeneous n-th order
differential equation at a singular point. Let $L$ be a linear
differential operator with coefficients analytic at zero.
If $L^*$ denotes the operator conjugate to $L$, then we will show
that the dimension of the kernel of $L$ is equal to the dimension
of the kernel of $L^*$. Certain representation theorems from
functional analysis will be used to describe the space of linear
functionals that contain the kernel of $L^*$. These results will
be used to derive a form of the Fredholm Alternative that will
establish a link between the solvability of $Ly = g$ at a singular
point and the kernel of $L^*$. The relationship between the roots
of the indicial equation associated with $Ly=0$ and the kernel of
$L^*$ will allow us to show that the kernel of $L^*$ is spanned
by a set of polynomials.
Submitted July 28, 2001. Published February 4, 2002.
Math Subject Classifications: 30A99, 34A30, 34M35, 46E15.
Key Words: linear differential equation; regular
singular point; analytic solution.