Representing Airy's function as a power series

In summary, the power series solution to d2y/dx2-xy=0 about x=-2 has the first five non-zero terms as y(-2) , y'(-2)(x+2), -y(-2)/2(x+2)^2, 0, and y(-2)/24(x+2)^4. A recurrence relation can be found to determine the remaining terms, but further analysis is needed to comment on the convergence of the series.
  • #1
Ratpigeon
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0

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?
 
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  • #2
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
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...
 

1. What is Airy's function?

Airy's function is a special mathematical function that appears in various areas of science, such as physics and engineering. It is named after the British astronomer and mathematician George Biddell Airy, who studied its properties in the 19th century.

2. How is Airy's function represented?

Airy's function can be represented in various forms, such as a power series, an integral, or a differential equation. In this case, we will focus on representing it as a power series.

3. What is a power series?

A power series is an infinite series of the form ∑n=0∞ an(x-a)n, where a is a fixed constant and an are coefficients. It is a useful mathematical tool for representing functions as a sum of polynomial terms, making it easier to manipulate and analyze them.

4. How is Airy's function represented as a power series?

Airy's function can be represented as a power series using its Taylor series expansion around the point x=0. This expansion is given by the formula: Ai(x) = ∑n=0∞ (-1)n (x/3)3n/2 / (2n n! Γ(3n/2 + 1)), where Γ is the gamma function.

5. What are the applications of representing Airy's function as a power series?

Representing Airy's function as a power series can help in solving various mathematical problems related to diffraction, wave propagation, and other physical phenomena. It can also be used in approximating the values of Airy's function for different values of x, which is useful in many engineering and scientific calculations.

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