(Elsevier, 2018) Ferreira González, Chelo; López García, José Luis; Pérez Sinusía, Ester; Matematika eta Informatika Ingeniaritza; Institute for Advanced Materials and Mathematics - INAMAT2; Ingeniería Matemática e Informática
We 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.