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Math Help - particular solution for non-homogeneous equation

  1. #1
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    particular solution for non-homogeneous equation

    hello, i just started learning about this and i'm a little bit confused on how to find the particular solution for these 2 non-homogeneous equations.

    S_n = S_{(n-1)}+2n and g_{n} = 5g_{(n-1)} - 6g_{(n-2)} + n + 5

    I know how to find a particular solution for an equation like g_{n} = g_{(n-1)} - 2g_{(n-2)} + 2 which is -1, but when there is a variable n instead of just a constant at the end, i get confused.

    can someone help me get started? thanks!
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  2. #2
    MHF Contributor chisigma's Avatar
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    Quote Originally Posted by break View Post
    hello, i just started learning about this and i'm a little bit confused on how to find the particular solution for these 2 non-homogeneous equations.

    S_n = S_{(n-1)}+2n and g_{n} = 5g_{(n-1)} - 6g_{(n-2)} + n + 5

    I know how to find a particular solution for an equation like g_{n} = g_{(n-1)} - 2g_{(n-2)} + 2 which is -1, but when there is a variable n instead of just a constant at the end, i get confused.

    can someone help me get started? thanks!
    The first recursive relation in 'standard form' is written as...

    s_{n+1}= s_{n} + 2\ (n+1) (1)

    If the 'initial condition' is specified as s_{0}=a , then the solution is easy to find...

    s_{1}= a + 2

    s_{1}= a + 2+4

    ...

    s_{n}= a + 2+4+...+2^{n}= a + 2\ (2^{n}-1) (1)

    ... and the 'particular solution' You are searching for is...

    \delta_{n}= 2\ (2^{n}-1) (2)

    Kind regards

    \chi \sigma
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