[math]y'' - xy = 0[/math]
I found two solutions to Airy's equation using Power Series.
One solution
[math]y_1(x) = a_0\sum_{n=0}^{\infty}\frac{1}{...9 \cdot 8 \cdot 6 \cdot 5 \cdot 3 \cdot 2} \ x^{3n}[/math]
As you can see in the denominator it is almost a factorial. I can't find a suitable formula for that. I also don't know how to find the value of [math]a_0[/math] to use this sum to approximate the values of Airy's function to some decimal places.
The second solution is similar with a little modification.
I found two solutions to Airy's equation using Power Series.
One solution
[math]y_1(x) = a_0\sum_{n=0}^{\infty}\frac{1}{...9 \cdot 8 \cdot 6 \cdot 5 \cdot 3 \cdot 2} \ x^{3n}[/math]
As you can see in the denominator it is almost a factorial. I can't find a suitable formula for that. I also don't know how to find the value of [math]a_0[/math] to use this sum to approximate the values of Airy's function to some decimal places.
The second solution is similar with a little modification.