Haile, Brian2020-07-132020-07-132002-02-04Haile, B. (2002). Analytic solutions of n-th order differential equations at a singular point. <i>Electronic Journal of Differential Equations, 2002</i>(12), pp. 1-14.1072-6691https://hdl.handle.net/10877/12052Necessary 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.Text14 pages1 file (.pdf)enAttribution 4.0 InternationalLinear differential equationRegular singular pointAnalytic solutionArticleThis work is licensed under a Creative Commons Attribution 4.0 International License.