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Representing Airy's function as a power series

  1. Apr 28, 2012 #1
    1. The problem statement, all variables and given/known data
    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

    2. Relevant equations

    ... calculus in general?
    and the taylor expansion of y(x) is - assuming remainder term is zero:
    [itex]\sum[/itex]y(n)(-2)/n! *(x+2)n (n from 0 to infinity)
    and y''(x) is:
    [itex]\sum[/itex]y(n)(-2)/(n-2)! *(x+2)n-2 (n from 2 to infinity
    3. The attempt at a solution
    I let x'=x+2, giving:
    [itex]\sum[/itex]y(n)(-2)/(n-2)! *(x')n-2-x'[itex]\sum[/itex]y(n)(-2)/n! *(x+2)n +2[itex]\sum[/itex]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?
     
  2. jcsd
  3. Apr 28, 2012 #2

    MathematicalPhysicist

    User Avatar
    Gold Member

    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.
     
  4. Apr 30, 2012 #3
    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...
     
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