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solving non-homogeneous difference equations Watch

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    I'm stuck on these questions and have been trying all day, please help!

    a)  u_n = 2^{n}u_{n-1} + \sqrt2^{n^2+2}, u_0 = 1

    I don't know how to find the characteristic polynomial for this, I get lambda = 1/2^n

    b)  u_n = 5u_{n-1} - 4u_{n-2} - 12n + 31, u_0 = 7, u_1 = 9

    for this, I did the characteristic polynomial: lambda = 1/4 and lambda = 1

     G(n) = A(1/4)^n + B.1^n 



P(n) = nM_0 + M_{1}n^2

    then when I try and subst. into the original, I get M_1 = 31/3 but M_0 cancels out.

    I know I need to find A and B

    c)  a_{n+2} = a_{n+1} - a_n + 21.2^n, a_0 = 10, a_1 = \frac{31+5\sqrt{3}}{2}

    I'm assuming this is similar to b) but what would P(n) be?

    Thanks
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    (Original post by ~tis)
    ...

    Thanks
    I can look at this tomorrow evening (bedtime soon) but would you be as kind as to post a photo of the question please
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    (Original post by TeeEm)
    I can look at this tomorrow evening (bedtime soon) but would you be as kind as to post a photo of the question please
    sorry just seen this
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    (Original post by ~tis)
    sorry just seen this
    I need to look up how to do the first one

    the second one I just did, bit messy but standard
    the "complimentary function" is A + B x 4n (no quarter)

    The third one I might try it tonight if I get a minute otherwise on Monday
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    (Original post by ~tis)
    sorry just seen this
    I just did the third one (also ok).
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    (Original post by TeeEm)
    I just did the third one (also ok).
    thanks going to try now.

    yeah, the first one's got me really stuck because of it being raised to n
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    (Original post by ~tis)
    thanks going to try now.

    yeah, the first one's got me really stuck because of it being raised to n
    I have written neat solutions to the last 2, so if you get stuck yell....

    The first one I will need to look up a few things before I attempt it which will now be on Monday

    all the best
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    (Original post by TeeEm)
    I have written neat solutions to the last 2, so if you get stuck yell....The first one I will need to look up a few things before I attempt it which will now be on Mondayall the best

    I still can't get the answer for b).

    I have G(n) = A + 4Bn and P(n) = nM_0 + M_1n^2

    Then do I subst. P(n) into the original equation?
    i.e.  nM_0 + M_{1}n^2 - 5nM_0 - 5M_{1}(n^{2} + 1) + 4nM_0 + 4M_{1}(n^{2}+2) = 31 - 12n

    but then I get M_1 = 31/3

    and I don't know what to do after this
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    (Original post by ~tis)
    I still can't get the answer for b).

    I have G(n) = A + 4Bn and P(n) = nM_0 + M_1n^2

    Then do I subst. P(n) into the original equation?
    i.e.  nM_0 + M_{1}n^2 - 5nM_0 - 5M_{1}(n^{2} + 1) + 4nM_0 + 4M_{1}(n^{2}+2) = 31 - 12n

    but then I get M_1 = 31/3

    and I don't know what to do after this
    I just finished teaching
    Give me a while to eat and I will post (b)
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    here is question 2
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    (Original post by TeeEm)
    here is question 2
    thank you
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    (Original post by ~tis)
    thank you
    no worries...
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    here is Q3
    I must remember to look at Q1
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