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Math Help - More Difference Equations

  1. #1
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    More Difference Equations

    I thought id post this in a new thread, i got the answer to the last one, but am kinda unsure of what to do in this -

    For each of rhe following difference equations find:
    (i) the general solutions
    (ii) the solution corresponding to the given initial conditions


    (A) <br />
x_{n} - 3x_{n-1}  = 2.3^n + n, x_{0} = 1/4<br />

    I am able to get Yn, but im confused with the rest of the question

    Also, with this question i am unsure of what to do with the '4', should i represent it as C?


    For each of rhe following difference equations find:
    (i) the general solutions
    (ii) the solution corresponding to the given initial conditions

    (B) <br />
x_{n+2} - 2x_{n+1} + x_{n} = 4, x_{0} = 3, x_{1} = 6<br />

    Thanks everybody!
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  2. #2
    Senior Member Peritus's Avatar
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    <br />
x_n  - 3x_{n - 1}  = 2.3^n  + n<br /> <br />

    solve the homogeneous equation, using the method I've already shown you.

    Then assume a particular solution of the following form:

    <br />
x_p  = C_1 2.3^n  + C_2 n<br />

    and apply the initial conditions...

    (B) x_{n + 2}  - 2x_{n + 1}  + x_n  = 4

    we know that the difference equation should hold for any positive n, in particular for n=n+1:

    x_{n + 3}  - 2x_{n + 2}  + x_{n + 1}  = 4<br />

    subtract the two equations:

    <br />
x_{n + 3}  - 3x_{n + 2}  + 3x_{n + 1}  - x_n  = 0<br /> <br />

    and this is a homogeneous equation you've already know how to solve...
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  3. #3
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    For the second question, i resulted in  (x-1)^3 is that right?
    And then i used  A(1)^n + Bn(1)^n + Cn^2(1)^n .. does that seem right?
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  4. #4
    Senior Member Peritus's Avatar
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    Right
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  5. #5
    Senior Member Peritus's Avatar
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    Please read the forum rules:

    15)Create a new thread for each new problem. Do not add new problems to existing threads especially when they are not your own thread.
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  6. #6
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    This is how i did the following question, but i seem to get stuck -

     x_{n} = 3x_{n-1} + 2.3^n + n, x_{0} = 1/4

    y_{n} = A3^n

    z_{n} = Bn3^n + Cn

    Bn3^n + Cn - 3(B(n-1)3^{n-1} + C(n-1)) = 2.3^n + n

    B3^n - 2Cn + 3C = 2.3^n + n

    B=2

    How do i obtain C, is that right so far?
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