Electron. J. Diff. Eqns., Vol. 2002(2002), No. 12, pp. 114.
Analytic solutions of nth order differential equations at a
singular point
Brian Haile
Abstract:
Necessary and sufficient conditions are be given for the
existence of analytic solutions of the nonhomogeneous nth order
differential equation at a singular point. Let
be a linear
differential operator with coefficients analytic at zero.
If
denotes the operator conjugate to
,
then we will show
that the dimension of the kernel of
is equal to the dimension
of the kernel of
.
Certain representation theorems from
functional analysis will be used to describe the space of linear
functionals that contain the kernel of
.
These results will
be used to derive a form of the Fredholm Alternative that will
establish a link between the solvability of
at a singular
point and the kernel of
.
The relationship between the roots
of the indicial equation associated with
and the kernel of
will allow us to show that the kernel of
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
email: bhaile@mail.nwmissouri.edu 
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