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Math Help - Non-homogeneous recurrence relations

  1. #1
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    Thumbs up Non-homogeneous recurrence relations

    Hello everyone

    I really need help on this work:

    Solve 4an+2 - 4an+1 + an = (1/3)^n

    Thanks in anticipation
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  2. #2
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    Quote Originally Posted by Rmathk View Post
    Hello everyone

    I really need help on this work:

    Solve 4an+2 - 4an+1 + an = (1/3)^n

    Thanks in anticipation
    If a_n is the general solution to the associated homogeneous equation, 4a_{n+2}- 4a_{n+1}+ a_n= 0, and b_n is any solution to the entire equation, 4a_{n+2}- 4a_{n+1}+ a_n= (\frac{1}{3})^n, then a_n+ b_n is the general solution to the entire equation.

    Do you know how to find the general solution to the associated homogeneous equation?

    To find a solution to the entire equation, try a particular solution b_n= A\left(\frac{1}{3}\right)^n.
    Last edited by CaptainBlack; October 24th 2009 at 11:53 PM. Reason: fixed LaTeX and added a bit of spoon feeding
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  3. #3
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    Quote Originally Posted by HallsofIvy View Post
    If a_n is the general solution to the associated homogeneous equation, 4a_{n+2}- 4a_{n+1}+ a_n= 0, and b_n is any solution to the entire equation, 4a_{n+2}- 4a_{n+1}+ a_n= (\frac{1}{3})^n, then a_n+ b_n is the general solution to the entire equation.

    Do you know how to find the general solution to the associated homogeneous equation?

    To find a solution to the entire equation, try [tex]a_n= A\left(\frac{1}{3}\right)^n[/itex].
    Thanks for your reply HallsofIvy

    I obtained the characteristic equation of the homogeneous equation as
    4(x^2) – 4x + 1 = 0
    x = (1/2)

    I suppose the general solution is then:
    an = (a+bn)[(1/2)^(n)]

    How do i find a solution to the entire equation? Am stuck
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  4. #4
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    Quote Originally Posted by Rmathk View Post
    Thanks for your reply HallsofIvy

    I obtained the characteristic equation of the homogeneous equation as
    4(x^2) 4x + 1 = 0
    x = (1/2)

    I suppose the general solution is then:
    an = (a+bn)[(1/2)^(n)]

    How do i find a solution to the entire equation? Am stuck
    go back and read HallsofIvy's post again

    CB
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