%0 Journal Article
%T On Series Solutions of y''- f(x)y=0 and Applications
%A P. N. Shivakumar and Yang Zhang
%J Advances in Difference Equations
%D 2013
%I 
%R 10.1186/1687-1847-2013-47
%X In this paper, we develop explicit expressions for the Taylor series coefficients in the formal Taylor series solution of the second order linear differential equation $y'' -f(x)y = 0$ for a given arbitrary function $f(x)$ in terms of initial conditions. As applications we apply our results to $f(x) = e^x, \cos(x)$ and Airy's equation, and give explicit formulas for the Taylor coefficients of $\exp(\sum_{k=0}^\infty a_{k}x^k)$ which is a long standing question.
%U http://www.advancesindifferenceequations.com/content/2013/1/47/abstract