On series solutions of $y'' - f(x)y=0$ and applications
Date
2013-03-04
Authors
Shivakumar, Pappur N
Zhang, Yang
Journal Title
Journal ISSN
Volume Title
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Abstract
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.
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Citation
Advances in Difference Equations. 2013 Mar 04;2013(1):47