Analysis of singular one-dimensional linear boundary value problems using two-point Taylor expansions

We consider the second-order linear differential equation (x(2) - 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, g and h are analytic in a Cassini disk D-r with foci at x = +/- 1 containing the interval [-1,1]. Then, the two end points of the interval may be regular singular points 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 exist.