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Non Homogeneous Recurrence Relation

  1. May 1, 2008 #1
    1. solve the following recurrence relation for an



    2. (n+2)an+1= 2(n+1)an+2[tex]^{n}[/tex], n>=0, a0=1
    I shifted the index, multiplied through by the 2[tex]^{n}[/tex] term and then subtracted the resulting equation from the original equation to get rid of the 2[tex]^{n}[/tex] term...


    3. I have gotten to this point
    (n+1)an-4(n)an-14(n-1)an-2=0


    I'm not really sure how to handle the (n+1), n, or (n-1) terms when looking for the particular/ homogeneous solution parts.
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
  3. May 3, 2008 #2

    epenguin

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    Homework Helper
    Gold Member

    can you see a way to change your variable a(n) that would do it? There is something consistemt between the successive terms.

    (You have missed a + out of your formula BTW.)
     
  4. May 3, 2008 #3
    well, if I sub in bm = (n+1)an into the original equation of
    (n+2)an+1 = 2(n+1)an+2n
    I get
    bm+1=2bm+2m
    (1) bm+1-2bm=2m
    (2) bm-2bm-1=2m-1
    (3) 2bm-4bm-1=2m
    (1)-(3)
    (4) bm+1-4bm+4bm-1=0
    (5) bm-4bm-1+4bm-2=0
    (6) r2-4r+4=0
    (7) (r-2)(r-2)=0
    (8) bm= c12m+c2m2m

    but I don't really know where to go from there?
    do I sub back in, or is there a way to use a0=1 with bm?
    I'm getting the feeling that I'm dong something wrong....
     
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