The theory of series solutions for second-order linear homogeneous ordinary differential equation is developed ab initio, using an elementary complex integral expression (based on Herrera' work [3]) derived and applied in previous papers [8,9]. As well as reproducing the usual expression for the recurrence relations for second-order equations, the general solution method is straight-forward to apply as an algorithm on its own, with the integral algorithm replacing the manipulation of power series by reducing the task of finding a series solution for second-order equations to the solution, instead, of a system of uncoupled simple equations in a single unknown. The integral algorithm also simplifies the construction of 'logarithmic solutions' to second-order Fuchs, equations. Examples, from the general science and mathematics literature, are presented throughout.Mathematics Subject Classification: 30B10, 30E20 34A25, 34A30