Regularity, synthesis, rigidity and analytic classification for linear ordinary differential equations of second order

We study second order linear homogeneous differential equations a(x)y'' + b(x)y' + c(x)y = 0 with analytic coefficients in a neighborhood of a regular singularity in the sense of Frobenius. These equations are model for a number of natural phenomena in sciences and applications in engineering. We address questions which can be divided in the following groups: (i) Regularity of solutions. (ii) Analytic classification of the differential equation. (iii) Formal and differentiable rigidity. (iv) Synthesis and uniqueness of ODEs with a prescribed solution. Our approach is inspired by elements from analytic theory of singularities and complex foliations, adapted to this framework. Our results also reinforce the connection between classical methods in second order analytic ODEs and (geometric) theory of singularities. Our results, though of a clear theoretical content, are important in justifying many procedures in the solution of such equations.