Geodesic
Analytic solutions of \(n\)-th order differential equations at a singular point
Electronic journal of differential equations, Tome 2002 (2002)
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.
Classification : 30A99, 34A30, 34M35, 46E15
Keywords: linear differential equation, regular singular point, analytic solution 
@article{EJDE_2002__2002__a76,
     author = {Haile,  Brian},
     title = {Analytic solutions of \(n\)-th order differential equations at a singular point},
     journal = {Electronic journal of differential equations},
     year = {2002},
     volume = {2002},
     zbl = {1004.34080},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/EJDE_2002__2002__a76/}
}
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%J Electronic journal of differential equations
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Haile,  Brian. Analytic solutions of \(n\)-th order differential equations at a singular point. Electronic journal of differential equations, Tome 2002 (2002). http://geodesic.mathdoc.fr/item/EJDE_2002__2002__a76/