Power Series Method to Solve Airy’s Differential Equation [Duplicate]

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 $a's$.