On series solutions of $y'' - f(x)y=0$ and applications
Shivakumar, Pappur N
AbstractIn 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 ( ∑ k = 0 ∞ a k x k ) , which is a long-standing question.
Advances in Difference Equations. 2013 Mar 04;2013(1):47