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Math Help - Difference Equations

  1. #1
    Junior Member
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    Difference Equations

    The question states -

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

    (c) x_{n+2} - 6x_{n+1} + 8x_{n} = 3.5^n, x_{0} = 3, x_{1} = 6

    Thanks for the help.. im not too sure what to do.. so any help wouldbe appreciated
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  2. #2
    Senior Member Peritus's Avatar
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    first we solve the homogeneous difference equation:

    <br />
x_{n + 2}  - 6x_{n + 1}  + 8x_n  = 0<br /> <br />

    we assume the solution is of the following form:

    <br />
r^n <br />

    we now plug in this expression into the DE:

    \begin{gathered}<br />
  r^{n + 2}  - 6r^{n + 1}  + 8r^n  = 0 \hfill \\<br />
   \hfill \\<br />
   \Leftrightarrow r^n \left( {r^2  - 6r + 8} \right) = 0 \hfill \\ <br />
\end{gathered}

    and solve for r:

    r = 4, 2 thus:

    <br />
x_h  = C_1 2^n  + C_2 4^n <br />

    now to find the particular solution we use the method of undetermined coefficients namely we assume that the solution has the form of the inhomogeneous term:

    <br />
x_p  = D3.5^n <br /> <br />

    we proceed to substituting the particular solution in the DE:

    <br />
\begin{gathered}<br />
  D3.5^{n + 2}  - 6D3.5^{n + 1}  + 8D3.5^n  = 3.5^n  \hfill \\<br />
   \hfill \\<br />
   \Leftrightarrow D3.5^n \left( {3.5^2  - 6 \cdot 3.5 + 8} \right) = 3.5^n  \hfill \\<br />
   \hfill \\<br />
   \Leftrightarrow D = \frac{1}<br />
{{3.5^2  - 6 \cdot 3.5 + 8}} \hfill \\ <br />
\end{gathered} <br /> <br />

    so the general solution is:

    <br />
x = x_h  + x_p  = C_1 2^n  + C_2 4^n  + D3.5^n <br />

    now all you have to do is apply the initial conditions to find the constants...
    Last edited by Peritus; January 25th 2008 at 01:10 PM.
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  3. #3
    Junior Member
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    i was able to do that bit by simply factorising the orginial equation, whilst treating it as a quadratic, thanks for showing me a new method though. How do i then find the constants?
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  4. #4
    Senior Member Peritus's Avatar
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    well I've already told you: apply the initial conditions:

    you're told that: <br />
x_0  = 3<br />

    which means that:
    <br />
x[n = 0] = 3<br /> <br />

    thus: <br />
x_0  = C_1 2^0  + C_2 4^0  + D3.5^0  = C_1  + C_2  + D = 3<br /> <br />

    after applying the second initial condition you'll get another equation ....
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  5. #5
    Junior Member
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    I wrote down the wrong initial conditions, but i got the answer eventually!.. Thanks for the help
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