Abstract
The Ministerio de Economia y Competitividad (project MTM2017-83490-P) and Gobierno de Aragon (project E24_17R) are acknowledged by their financial support.
Highlights
In [6] we considered the second-order linear equation y + f (x)y + g(x)y = h(x) in the interval (−1, 1) with initial conditions or boundary conditions of the type Dirichlet, Neumann or mixed Dirichlet–Neumann
The two-point Taylor expansion of the solution y(x) at the end points ±1 was used to give a criterion for the existence and uniqueness of analytic solutions of the initial or boundary value problem and approximate the solutions when they exist
Where f, g and h are analytic in a Cassini disk with foci at x = ±1 containing the interval [−1, 1], α, β ∈ C and B is a 2 × 4 matrix of rank two which defines the initial conditions or the boundary conditions (Dirichlet, Neumann or mixed)
Summary
The two-point Taylor expansion of the solution y(x) at the end points ±1 was used to give a criterion for the existence and uniqueness of analytic solutions of the initial or boundary value problem and approximate the solutions when they exist. In [6] we improved the ideas of the previous paragraph for the regular case (when f (−1) = g(−1) = h(−1) = 0) using, not the standard Taylor expansion in the associated initial value problem (1.2), but a two-point Taylor expansion [4] at the end points x = ±1 directly in the differential equation and in the boundary conditions. We use the two-point Taylor expansion of the solution y(x) to give a criterion for the existence and uniqueness of analytic solutions based on the data of the problem, not based on the knowledge of the general solution of the differential equation. The analysis of this paper paper follows the same pattern as the analysis of [5]
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