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Find general solution from recurrence relation, done most of work

  1. Jul 15, 2008 #1
    1. The problem statement, all variables and given/known data
    2xy" + y' + xy =0 xo=0
    find the indicial equation, recurrence relation and series solution.


    2. Relevant equations



    3. The attempt at a solution
    I've done most of the work but i don't know how to use a recurrence relation to obtain a y1 and y2 series solution.
    this is what i've done:

    xy= sum (n=2 to infinity) an-2 x^(r+n-1)
    y'= sum(n=1 to infinity) (r+n) an x^(r+n-1)
    2xy''= 2*sum (n=2 to infinity) (r+n) (r+n-1) an x^(r+n-1)

    indicial equation:
    rao + 2r(r-1) ao=0
    r+ 2r^2-2 = 0
    r= 0 , r=1/2

    {(r+n) (r+n-1) an + (r+n) an + an-2}=0
    an= - an-2/ (r+n)(r+n-1)

    r=0:
    ao=1 a1=0
    a2= a0/2*1
    a3= a1/ 3*2
    a4= a2/4*3
    a5= a3/5*4

    what kind of pattern could i get out of this? the denominator looks like an n! but not exactly. Where can i go from here to get a y1 ?
     
  2. jcsd
  3. Jul 15, 2008 #2

    nicksauce

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    It seems to be that it should be easy to show, unless I'm mistaken, that
    a_{odd} = 0
    a_{even} = 1/n!
     
  4. Jul 16, 2008 #3

    HallsofIvy

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    The first thing you should have done is write out a few terms explicitely.

    If a0= 1 then a2= a0/2*1= 1/2. If a1= 0, a3= a1/3*2= 0. a4= a2/(4*3)= (1/2)1/(3*4))= 1/(2*3*4), a5= a3/(5*4)= 0/(5*4)= 0, a6= a4/(6*5)= [1/(2*3*4)][1/(6*5)]= 1/(2*3*4*5*6).

    Do you see now?
     
  5. Jul 16, 2008 #4
    yes I do, thank you very much. So to get y2 all i have to do is change from r=0 to r=1/2 and keep ao=1 and a1=0 right?
     
  6. Jul 17, 2008 #5

    HallsofIvy

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    Yes.
     
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