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

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.