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.