A novel analytical operator method to solve linear ordinary differential equations with variable coefficients

TL;DR

This paper introduces a new analytical operator method for solving linear ordinary differential equations with variable coefficients, providing explicit solutions in series and integral forms without recurrence or Green's functions.

Contribution

The method offers a novel approach to solving linear ODEs with singularities, yielding exact solutions directly in series and integral forms, simplifying the process compared to traditional techniques.

Findings

Solutions in analytic series form obtained without recurrence relations.

Exact solutions in Mellin-Barnes-type contour integrals provided.

Closed-form solutions for special functions in physics derived.

Abstract

A new analytical operator method is discussed which solves linear ordinary differential equations with regular singularities. Solutions are obtained in analytic series form and also in Mellin-Barnes-type contour integral form. Exact series solution is obtained without having to calculate series coefficients by recurrence relation.Both homogeneous and inhomogeneous equations are solved identically without having to calculate the Green's function explicitly in the case of inhomogeneous equation.Closed-form solutions are obtained for all the special functions appearing in mathematical physics. For a second-order equation both the independent solutions are obtained without invoking Wronskians, even when the indices differ by an integer.