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Science and engineering math: Difference equation, non-homogeneous

  1. Mar 18, 2012 #1
    1. The problem statement, all variables and given/known data

    Solve the difference equation: an+2 - 5an+1 + 6an = 4n
    Subject to a0 = 0 & a1 = 1

    2. Relevant equations

    an = Arn

    3. The attempt at a solution
    I got two solutions for the first part for when it is homogeneous by substituting an = Arn into the equation and solving for r.
    an = A(3)n
    an = A(2)n

    I just don't know where to start for non-homogeneous. I tried an = B*4n and I got B*4n+2 -5B*4n+1 + 6B*4n = 4n but I don't know where to go from there.
     
  2. jcsd
  3. Mar 18, 2012 #2

    tiny-tim

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    welcome to pf!

    hi chatterbug219! welcome to pf! :smile:
    erm :redface:

    4n+2/4n = … ? :wink:
     
  4. Mar 18, 2012 #3
    So B = 1/2
    And plugging that in...
    an = (1/2)(4n) for the particular solution
    Making the general solution:
    an = A(2)n + (1/2)(4n)
    Right?
     
  5. Mar 19, 2012 #4

    tiny-tim

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    hi chatterbug219! :smile:

    (just got up :zzz:)
    you mean an = A(2)n + B(3)n + (1/2)(4n) :wink:

    (or you could write it 2n(2n-1 + A) + 3nB )
     
  6. Mar 19, 2012 #5
    No...if you plug in the initial conditions they final answers don't match up...
    Only 2 works
     
  7. Mar 19, 2012 #6

    tiny-tim

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    yes, but to solve for the initial condition, you also need to find A
     
  8. Mar 19, 2012 #7

    Ray Vickson

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    The simplest way is to determine the generating function [itex]A(z) = \sum_{n=0}^{\infty} a_n z^n [/itex], then invert it to find [itex] \{ a_n \}. [/itex] Google "difference equations" to see hundreds of examples.

    RGV
     
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