doi:10.1016/j.jmaa.2018.03.041engFerreira, C.López, J.L.Pérez Sinusía, E.The use of two-point Taylor expansions in singular one-dimensional boundary value problems IART-2018-105253We consider the second-order linear differential equation (x+1)y¿+f(x)y'+g(x)y=h(x) in the interval (-1, 1) with initial conditions or boundary conditions (Dirichlet, Neumann or mixed Dirichlet–Neumann). The functions f(x), g(x) and h(x) are analytic in a Cassini disk Dr with foci at x=±1 containing the interval [-1, 1]. Then, the end point of the interval x=-1 may be a regular singular point of the differential equation. The two-point Taylor expansion of the solution y(x) at the end points ±1 is used to study the space of analytic solutions in Dr of the differential equation, and to give a criterion for the existence and uniqueness of analytic solutions of the boundary value problem. This method is constructive and provides the two-point Taylor approximation of the analytic solutions when they exist.2018http://zaguan.unizar.es/record/7871010.1016/j.jmaa.2018.03.041http://zaguan.unizar.es/record/78710oai:zaguan.unizar.es:78710info:eu-repo/grantAgreement/ES/MINECO/MTM2014-52859-PJournal of Mathematical Analysis and Applications 463, 2 (2018), 708-725by-nc-ndhttp://creativecommons.org/licenses/by-nc-nd/3.0/es/info:eu-repo/semantics/openAccess