Representing Airy's function as a power series

[Ratpigeon]

Homework Statement

Find the first five non-zero terms of the power series solution to
d²y/dx²-xy=0 about x=-2; y(-2)=1;y'(-2)=1/2

Homework Equations

... calculus in general?
and the taylor expansion of y(x) is - assuming remainder term is zero:
$\sum$y⁽ⁿ⁾(-2)/n! *(x+2)ⁿ (n from 0 to infinity)
and y''(x) is:
$\sum$y⁽ⁿ⁾(-2)/(n-2)! *(x+2)^n-2 (n from 2 to infinity

The Attempt at a Solution

I let x'=x+2, giving:
$\sum$y⁽ⁿ⁾(-2)/(n-2)! *(x')^n-2-x'$\sum$y⁽ⁿ⁾(-2)/n! *(x+2)ⁿ +2$\sum$y⁽ⁿ⁾(-2)/n! *(x+2)ⁿ
(parameters of first sum are from 2 to infinity, of second two sums from 0 to infinity)
But I'm not sure what to do from here - do I need to find y?

 

[MathematicalPhysicist]

You know already two terms, namely: y(-2) , y'(-2)(x+2).

Now in order to find the other non-zero terms, just use your ODE, equate terms with the same powers of x^n to zero, and from there you can find a recurrence relation that will help you find the other terms.

Post back if your'e still stuck.

 

[Ratpigeon]

I get a recurrence relation:
a_(n+3)=[a_n-2a_(n+1)-1]/(n+3)(n+2)
but I don't think that can be right, because I need to comment on the convergence...