Solving ordinary differential equations (ODE) with irregular singularities in complex domains can be done either formally or analytically. At the formal level, a method by Poincaré allows to compute formal (divergent) solutions. Then, one can use Borel summation to "resum" the divergent series and construct an analytic solution. In this talk, I will describe a different approach that builds analytic solutions by turning the ODE into an integral equation of Volterra type. More precisely, I will build a solution that is Borel regular, namely it is asymptotic to the Poincaré solution and equal to its Borel sum. This is based on a joint project with A. Fenyes, arXiv:2407.01412.
