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Math Help - difference equation

  1. #1
    Junior Member
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    Feb 2009
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    difference equation

    I am very stuck as to how to attempt this question, would really appreciate some help as lecturer was not clear

    xn+6 - 3xn+4 + 3xn+2 - xn = 0

    given initial condition

    X0=X2=X4=0, X1=2, X3=12, X5=30

    Please help me, would be so grateful
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  2. #2
    Super Member

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    Lexington, MA (USA)
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    Hello, manalive04!

    I have an "eyeball" solution . . .


    x_{n+6} - 3x_{n+4} + 3x_{n+2} - x_n \;=\;0

    Given initial conditions:

    . . \begin{array}{c}x_0\:=\:0\\<br />
x_1 \:=\:2 \\ x_2 \:=\:0 \\ x_3 \:=\:12 \\ x_4 \:=\:0 \\ x_5 \:=\:30 \\<br />
\vdots \end{array}

    We have: . \begin{array}{ccc}x_0 &=&0 \\ x_1 &=&1\!\cdot\!2 \\ x_2 &=& 0 \\ x_3 &=&3\!\cdot\!4 \\ x_4&=& 0 \\ x_5 &=& 5\!\cdot\!6 \\ \vdots & &\vdots \end{array}


    We see that: . x_n \;=\;\begin{Bmatrix}0 & \text{for even }n \\ n(n+1) & \text{ for odd }n \end{Bmatrix}


    Therefore, the general term is: . x_n \;=\;\frac{1 - (-1)^n}{2}\,n(n+1)

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