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Math Help - Difference Equations

  1. #1
    Forum Admin topsquark's Avatar
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    Difference Equations

    I have been reading the recent difference equation post and a question occurred to me. (Please forgive me I know many techniques but little theory for linear algebra.)

    I know how to solve something like
    a_{n + 2} - 3a_n = 2

    But what about
    na_{n + 2} - 3a_n = 0
    for example. Is there a general method to attack something like this?

    (I don't know if this even has a "nice" solution, I'm just using it as a sample equation where one or more of the coefficients depends on n.)

    -Dan
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  2. #2
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    Particular cases are often easier than general ones. The example you listed in fact has a straightforward solution.

    a_{n+2} = \frac{3a_n}{n}. If you work relation for small n, you find the general formula for a as a_{2m} = \frac{3^m }{2^m m!}a_2, \ a_{2m-1} = \frac{3^m(2m)!}{2^mm!}a_1, \ m=1, 2, ... where a_1 and a_2 are arbitrary.
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  3. #3
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    Quote Originally Posted by topsquark View Post
    (I don't know if this even has a "nice" solution, I'm just using it as a sample equation where one or more of the coefficients depends on n.)
    I am moving this to Discrete Math.

    Think of Difference Equations as Differencial equations.

    Does a general differencial equation have a "nice" solutions? I wish!

    However, there is a method to finding them. A very powerful method is known as generating functions.

    You can learn more about them here.
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