Analytic Solutions of n-th Order Differential Equations at a Singular Point

Haile, Brian
Journal Title
Journal ISSN
Volume Title
Southwest Texas State University, Department of Mathematics
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 and the kernel of L* will allow us to show that the kernel of L* is spanned by a set of polynomials.
Linear differential equation, Regular singular point, Analytic solution
Haile, B. (2002). Analytic solutions of n-th order differential equations at a singular point. <i>Electronic Journal of Differential Equations, 2002</i>(12), pp. 1-14.