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Math Help - Equation in finite differences

  1. #1
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    May 2009
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    Equation in finite differences

    Hi all!

    Can someone help me with this exercise? I don't know how to start

    Calculate the equation in finite differences of:

    14x[n]-9x[n-1]+x[n-2]=70delta[n+3]-10delta[n+2]

    Thanks in advance
    Greetings
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  2. #2
    Senior Member
    Joined
    Apr 2009
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    Atlanta, GA
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    408

    Simple Substitution

    First, realize that \Delta_n=x_{n+1}-x_n by definition. So, substitute:

    14x_n-9x_{n-1}+x_{n-2}=70(x_{n+4}-x_{n+3})-10(x_{n+2}-x_{n+1})

    The way recursive formulas are defined is by relating x_n to a formula involving several values x_k for k<n, so we want to isolate the x with the highest subscript, in this case, x_{n+4}.

    x_{n+4}=\frac{1}{70}(80x_{n+3}-10x_{n+2}+0x_{n+1}+14x_n-9x_{n-1}+x_{n-2})

    More formally,

    x_{n+6}=\frac{1}{70}(80x_{n+5}-10x_{n+4}+0x_{n+3}+14x_{n+2}-9x_{n+1}+x_n) for all n\in\mathbb{N}

    Therefore, to actually find the terms in this sequence, we would need to know the first six values of the sequence.
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