Answer of differential equation near point t = 0

Question

I have this:

$$y''-ty=0, \\ y(0)=1, y'(0)=0$$

The question is asking me to find the answer on point $t = 0$

What I've tried so far is

$$y=\sum_{n=0}^{\infty}a_n t^n \\$$

$$\sum n(n-1)a_n t^{n-2} + \sum a_nt ^ {n + 1} $$

and expanding on this I reach: $$a_{n+3}=\dfrac {a_n}{n^2+5n+6}$$

but I'm stuck here. Any ideas?

Answer 1

This is an Airy equation that has solutions in Airy functions that are defined as your power series solution and its complementary basis solution.

What you have to determine is that $a_2=0$ and thus all $a_{2+3k}=0$. You could try to find more compact formulas for the $a_{3k}$ and $a_{3k+1}$ subsequences using gamma functions or similar expressions for products of segments of arithmetic sequences.