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Math Help - Solve The Difference Equation

  1. #1
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    Solve The Difference Equation

    f_{n+2} - 2f_{n+1}+ 5f_n = 0

    given that f_0=1 and f_1=3

    Auxiliary equation:

    m^2 - 2m + 5 = 0

    Therefore:

    m = 1 + 2i or m = 1 - 2i

    What would the complementry fuction be and paticular solution to work towards the solution of:

    f_n = [(1-i)/2][1+2i]^n + [(1-i)/2][1-2i]^n
    Last edited by CaptainBlack; April 29th 2009 at 09:43 PM.
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  2. #2
    Grand Panjandrum
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    Quote Originally Posted by louboutinlover View Post
    f_{n+2} - 2f_{n+1}+ 5f_n = 0

    given that f_0=1 and f_1=3

    Auxiliary equation:

    m^2 - 2m + 5 = 0
    Which is derived from an assumed solution of the form f_n=m^n

    Therefore:

    m = 1 + 2i or m = 1 - 2i
    OK, so you have a general solution of the form:

    f_n=A (1+2\text{i})^n+B(1-2\text{i})^n

    and the initial conditions will allow you to determine A and B, which you appear to have done.

    What would the complementry fuction be and paticular solution to work towards the solution of:

    f_n = [(1-i)/2][1+2i]^n + [(1-i)/2][1-2i]^n
    What are you asking here?

    CB
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  3. #3
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    So using the initial conditions would give:

    A+B=1

    A(1+2i)+B(1-2i)=3

    But then how do you solve these equations ?
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  4. #4
    Grand Panjandrum
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    Quote Originally Posted by louboutinlover View Post
    So using the initial conditions would give:

    A+B=1

    A(1+2i)+B(1-2i)=3

    But then how do you solve these equations ?
    A=1-B from 1st,

    (1-B)(1+2\text{i})+B(1-2\text{i})=3 by substituting from the above into the second.

    So:

     <br />
B[-(1+2\text{i})+(1-2\text{i})]=3-(1+2\text{i})<br />

     <br />
B=\frac{-1+1\text{i}}{2\text{i}}=\frac{1+i}{2}<br />

    Assuming my algebra is right (check it, it appears to differ from what you are give by a sign here or there).

    CB
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