TITLE: Power series method to solve Airy’s differential equation
QUESTION [3 upvotes]: Using power series method, solve Airy’s equation 
$$y′′+ xy = 0.$$
How do I start solving this? Thanks in advance!

REPLY [2 votes]: Given:
$$\tag 1 y''+ x y = 0$$
Solve this using Power Series.
We assume:
$$y = \sum_{m=0}^\infty a_mx^m$$
Thus we have:
$$y'' = \sum_{m=2}^\infty m(m-1)a_mx^{m-2}$$
Substituting into the $(1)$, we get:
$$\sum_{m=2}^\infty m(m-1)a_mx^{m-2} + \sum_{m=0}^\infty a_mx^{m+1} = 0$$
Aligning starting points for the series, we have:
$$2a_2 + \sum_{m=1}^\infty ((m+1)(m+2)a_{m+2} + a_{m-1})x^m = 0$$
Now equate terms to zero and solve for the $a's$.
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