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Representing Airy's function as a power series

  • Thread starter Ratpigeon
  • Start date
  • #1
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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:
[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

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?
 

Answers and Replies

  • #2
MathematicalPhysicist
Gold Member
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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.
 
  • #3
56
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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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