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On Series Solutions of y''- f(x)y=0 and Applications

DOI: 10.1186/1687-1847-2013-47

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Abstract:

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.

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