Two-point Taylor expansions and one-dimensional boundary value problems

We consider second-order linear differential equations φ(x)y′′ + f(x)y′ + g(x)y = h(x) in the interval (−1, 1) with Dirichlet, Neumann or mixed Dirichlet-Neumann boundary conditions. We consider φ(x), f(x), g(x) and h(x) analytic in a Cassini disk with foci at x = ±1 containing the interval (−1, 1). The two-point Taylor expansion of the solution y(x) at the extreme points ±1 is used to give a criterion for the existence and uniqueness of solution of the boundary value problem. This method is constructive and provides the two-point Taylor approximation of the solution(s) when it exists.