On the procedure for the series solution of second-order homogeneous linear differential equation via the complex integration method

Abstract

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

Some remarks on the differential transform method

“…Interestingly, the Taylor-Cauchy transform [5,19] the differential transform [8][9][10][11][12]20] and the complex integral method [2] were all developed as techniques for the solution of nonlinear differential equations. However, as has been shown in, for example [13][14][15][16][17], such methods have exceptional efficacy in the solution of linear differential equations also.…”

“…As Bervillier again points out [1], the differential transform method has played a crucial role, as it has been developed, in the solution of wider and wider classes of differential equations, even fractional differential equations. Such generalizations are mirrored in the development of the complex integral method [13][14][15][16][17]. For example in reference [15] the complex integral method is extended to the solution of partial differential equations including systems of equations and Frobenius series solutions of partial differential equations.…”