Electron. J. Diff. Eqns., Vol. 2002(2002), No. 12, pp. 1-14.

Analytic solutions of n-th order differential equations at a singular point

Brian Haile

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.

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Brian Haile
Department of Mathematics and Statistics
Northwest Missouri state university
800 University Drive
Maryville, MO 64468 USA
e-mail: bhaile@mail.nwmissouri.edu

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