The use of two-point Taylor expansions in singular one-dimensional boundary value problems I

We consider the second-order linear differential equation (x+1)y"1+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 D-r. 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 D-r. 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 odst. (C) 2018 Elsevier Inc. All rights reserved.