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.