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Theoretical/non-tedious question about power series solution of y'' + y = 0

  1. Apr 14, 2012 #1

    s3a

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    1. "The problem statement, all variables and given/known data
    Find a recurrence formula for the power series solution around x = 0 for the differential equation given in the previous problem."

    The previous problem says:
    "Determine whether x = 0 is an ordinary point of the differential equation y'' + y = 0."


    2. Relevant equations
    Power series and related stuff.

    3. The attempt at a solution
    I have the solutions for both of these problems and I also know how to do them both. My question is just:

    If x = 0 was not an ordinary point, what would that mean? Would that mean that I cannot assume a power series solution of the form y = [n=0 to inf] Σ[a_n (x - x_0)^n] (where x_0 = 0 in this case) exists or what?

    Any input would be greatly appreciated!
    Thanks in advance!
     
  2. jcsd
  3. Apr 14, 2012 #2

    LCKurtz

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    If ##x=0## is a regular singular point, your series has to look like$$
    \sum_{n=0}^\infty a_nx^{n+r}$$
     
  4. Apr 18, 2012 #3

    s3a

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    Thanks for your answer. I have a bit more to ask though.

    It seems that I can assume there exists a solution y = Σ from n = 0 to inf of a_n * x^n if x = 0 is an ordinary point. I asked what would happen if x = 0 were not an ordinary point and your answer was good for me but now I want to know what if the question asked me to find the power series solution about x = {any one chosen finite real} and if x = {that same one chosen finite real} is an ordinary point, then would that still mean that a solution of the form y = Σ from n = 0 to inf of a_n * x^n exists?

    Also does it have to strictly be a real? I'm not looking to go too deep because my course doesn't require it but I just want to know if the answer is a yes or no since I plan to study this further for fun when I am done school.
     
  5. Apr 18, 2012 #4

    LCKurtz

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    If ##x=a## is an ordinary point and you want the expansion around that point, you would use a Taylor series centered at ##a##:$$
    y=\sum_{n=0}^\infty c_n(x-a)^n$$
     
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