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Solving differential equation using the Power Series Method

  1. Dec 16, 2008 #1
    1. The problem statement, all variables and given/known data

    x²y'' + (x² - 2)y = 0

    2. Relevant equations

    N/A

    3. The attempt at a solution

    I divided through by x² to get the equation in standard form. Then I plugged in the power series representation for y and y'' into the equation and got to this point...

    [​IMG]

    Now, I know that I want the powers on x to be the same. But before I do an index shift, I'm not really sure how to handle the (x² - 2)/x² in front of the sum in the second term. If it were just an x, then I would be able to strip one element off the index. But since it's not just an x or power of x, I'm not really sure how to handle it.

    Thanks for any hints!
     
  2. jcsd
  3. Dec 16, 2008 #2
    Well

    [tex] \frac{x^{2} - 2}{x^{2}} = 1 - \frac{2}{x^{2}} [/tex]

    And then just distribute the sum?
     
  4. Dec 16, 2008 #3
    That's where I'm confused. I'm not sure how to distribute the sum with that.

    1 - 2/x2 = 1 - 2x-2 so does that mean the sum in the second term becomes xn-2? Not sure.......
     
  5. Dec 16, 2008 #4
    Well you will now have 2 sums...

    So if by the 2nd term you mean 2nd sum, then yes (with a 2 out front)
     
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