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Find the power series in x-x0?

  1. Jan 18, 2017 #1
    1. The problem statement, all variables and given/known data
    Find the power series in x-x0 for the general solution of y"-y=0; x0=3.
    2. Relevant equations
    None.

    3. The attempt at a solution
    Let me post my whole work:
     

    Attached Files:

  2. jcsd
  3. Jan 18, 2017 #2
    What I did was the substitution method using z=x-x0.
    The answer for this problem is Math2.jpg
     
  4. Jan 18, 2017 #3

    Ray Vickson

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    No, please don't. The PF standard is for you to type out the problem and the solution, reserving images for things like diagrams or geometric constructions, etc.

    I know that some helpers answer questions about handwritten solution images, but most will not bother. You should consult the pinned post "guidelines for Students and Helpers", by Vela, for a good explanation about this and similar issues.
     
  5. Jan 18, 2017 #4
    Can you please take a look at the work that I posted? It's clearly written.
     
  6. Jan 18, 2017 #5

    PeroK

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    It looks like the right answer in post #2, if that's all you are asking.
     
  7. Jan 18, 2017 #6
    I know that's the right answer, but what should I do to get to the right answer after the last step in my work? That's where I got stucked.
     
  8. Jan 18, 2017 #7

    PeroK

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    You need to spot a pattern in the coefficients (and verify it by induction if need be). You can see from the answer that you need to separate ##n## even and odd.
     
  9. Jan 18, 2017 #8
    You mean this:

    n=2m (even index)
    a2m+2=a2m/[(2m+2)(2m+1)]
    ----------------------------------------------------------------------------
    n=2m+1 (odd index)
    a2m+3=a2m+1/[(2m+3)(2m+2)]
     
  10. Jan 18, 2017 #9

    PeroK

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    Yes, but can you see the pattern? The answer gives you a big clue!
     
  11. Jan 18, 2017 #10
    So how do I get to the answer? I know where x-3 comes from.
     
  12. Jan 18, 2017 #11

    PeroK

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    You get to the answer by noticing that ##1 \times 2 \times 3 \times \dots \times n = n!##

    The clue was that the answer has ##n!## in it.
     
  13. Jan 18, 2017 #12
    I still don't really get it.
     
  14. Jan 18, 2017 #13

    PeroK

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    You have:

    ##(n+2)(n+1)a_{n+2} = a_n##

    Hence:

    ##a_{n+2} = \frac{a_n}{(n+2)(n+1)}##

    For ##n## even this gives:

    ##a_2 = \frac{a_0}{2}, \ a_4 = \frac{a_2}{12} = \frac{a_0}{24}, \ a_6 = \frac{a_4}{30} = \frac{a_0}{720} \dots##

    And, now by insight, inspiration (or looking at the answer) you have to notice that ##2, 24, 720 \dots## are the even factorials and hence ##a_n = \frac{a_0}{n!}##

    Odd ##n## is much the same.
     
  15. Jan 18, 2017 #14
    I got it now. Thank you so much.
     
  16. Jan 18, 2017 #15

    berkeman

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    This thread is in the Homework Help forums...
     
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