Abstract
This chapter describes the series solution of second-order ordinary differential equations (ODEs) by the method of Frobenius, with application to ODEs of importance in physics and engineering. The series solution method of Frobenius is presented, together with the conditions under which it must succeed. This method is applied to the Legendre and associated Legendre equations, the Bessel equation, the Hermite equation, the Laguerre (and associated Laguerre) equations, and the Chebyshev (type 1) equation. The solutions are also characterized by their generating functions, their Rodrigues formulas, their recurrence formulas, and their use in orthogonal expansions. In addition to discussing solutions to the Bessel ODE, there are presentation of modified and spherical Bessel functions. All the topics of this chapter are treated using symbolic computation and include graphs showing the essential features of these function sets.
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