- #1
Destroxia
- 204
- 7
1. Homework Statement
##x^{2}y'' + (x^{2} + 1/4)y=0##
3. The Attempt at a Solution
First I found the limits of a and b, which came out to be values of a = 0, and b = 1/4
then I composed an equation to solve for the roots:
##r^{2} - r + 1/4 = 0## ##r=1/2##
The roots didn't differ by an integer so the equation must take the form of
##y(x)= \sum_{n=0}^\infty a_{n}x^{n+(1/2)}##
##y'(x)= \sum_{n=1}^\infty (n+1/2)a_{n}x^{n-(1/2)}##
##y''(x)= \sum_{n=2}^\infty 1/4(2n-1)(2n+1)a_{n}x^{n-3/2}##
Now I plugged the corresponding derivatives into the differential equation:
## x^{2} \sum_{n=2}^\infty 1/4(2n-1)(2n+1)a_{n}x^{n-3/2} + (x^{2} + 1/4) \sum_{n=0}^\infty a_{n}x^{n+(1/2)} = 0##
Then, I distributed the x terms through the series
## \sum_{n=2}^\infty 1/4(2n-1)(2n+1)a_{n}x^{n+1/2} + \sum_{n=2}^\infty a_{n-2}x^{n+(1/2)} +\sum_{n=0}^\infty (1/4)a_{n}x^{n+(1/2)}##
Now I pull out 2 terms from the \sum_{n=0}^\infty a_{n}x^{n+(1/2)} term:
## \sum_{n=2}^\infty 1/4(2n-1)(2n+1)a_{n}x^{n+1/2} + \sum_{n=2}^\infty a_{n-2}x^{n+(1/2)} + (1/4)a_{0}x^{1/2} +(1/4)a_{1}x^{3/2} + \sum_{n=2}^\infty (1/4)a_{n}x^{n+(1/2)}##
Then of course I would find the recursive formula, but I just wanted to make sure this is the proper way to set everything up before I proceed with that part of the problem, as I always have issues with these series solutions.
##x^{2}y'' + (x^{2} + 1/4)y=0##
3. The Attempt at a Solution
First I found the limits of a and b, which came out to be values of a = 0, and b = 1/4
then I composed an equation to solve for the roots:
##r^{2} - r + 1/4 = 0## ##r=1/2##
The roots didn't differ by an integer so the equation must take the form of
##y(x)= \sum_{n=0}^\infty a_{n}x^{n+(1/2)}##
##y'(x)= \sum_{n=1}^\infty (n+1/2)a_{n}x^{n-(1/2)}##
##y''(x)= \sum_{n=2}^\infty 1/4(2n-1)(2n+1)a_{n}x^{n-3/2}##
Now I plugged the corresponding derivatives into the differential equation:
## x^{2} \sum_{n=2}^\infty 1/4(2n-1)(2n+1)a_{n}x^{n-3/2} + (x^{2} + 1/4) \sum_{n=0}^\infty a_{n}x^{n+(1/2)} = 0##
Then, I distributed the x terms through the series
## \sum_{n=2}^\infty 1/4(2n-1)(2n+1)a_{n}x^{n+1/2} + \sum_{n=2}^\infty a_{n-2}x^{n+(1/2)} +\sum_{n=0}^\infty (1/4)a_{n}x^{n+(1/2)}##
Now I pull out 2 terms from the \sum_{n=0}^\infty a_{n}x^{n+(1/2)} term:
## \sum_{n=2}^\infty 1/4(2n-1)(2n+1)a_{n}x^{n+1/2} + \sum_{n=2}^\infty a_{n-2}x^{n+(1/2)} + (1/4)a_{0}x^{1/2} +(1/4)a_{1}x^{3/2} + \sum_{n=2}^\infty (1/4)a_{n}x^{n+(1/2)}##
Then of course I would find the recursive formula, but I just wanted to make sure this is the proper way to set everything up before I proceed with that part of the problem, as I always have issues with these series solutions.