86 - Series Solution*

Abstract

This chapter discusses series solution applicable to homogeneous linear ordinary differential equations. If an infinite series is substituted into a linear equation, the different coefficients may be matched to obtain recurrences for the coefficients of the series; solving these recurrences results in an explicit solution. In the procedure, for a homogeneous linear second-order ordinary differential equation in the form y“ + P(x)y' + Q(x)y = 0, a series solution is searched around the point x = 0. An expansion about any other point, x0, could be determined by changing the independent variable to t = x − x0, and then analyzing the resulting equation near t = 0. The chapter highlights that when the given linear ordinary differential equation has an irregular singular point, then series solutions are difficult to obtain and they may be slowly convergent. Often, the WKB method is used to approximate the solution near an irregular singular point.