Analytic Solutions of n-th Order Differential Equations at a Singular Point
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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.
CitationHaile, B. (2002). Analytic solutions of n-th order differential equations at a singular point. Electronic Journal of Differential Equations, 2002(12), pp. 1-14.
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