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Thread: 3rd Order Recurrence Relation

  1. #1
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    3rd Order Recurrence Relation

    Hi,

    I have a 3rd Order Homogenous Recurrence Relation, with a characteristic polynomial of

    x^3 - r1x^2 - r2x - r3. This polynomial has three roots, \lambda1, \lambda2, \lambda3, which are the same.

    I know for a 2nd Order Recurrence Relation with equal roots, the general solution of the relation is C1\lambda^n + C2n\lambda^n, for some constants C1, C2, but what is the general solution for a 3rd Order with equal roots?

    Thankyou.
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  3. #3
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    Hello, jsbach!

    I have a 3rd Order Homogenous Recurrence Relation,
    . . with a characteristic polynomial of: .$\displaystyle x^3$ - $\displaystyle r_1$$\displaystyle x^2$ - $\displaystyle r_2$$\displaystyle x $ - $\displaystyle r_3$

    This polynomial has three roots: $\displaystyle \lambda_1$, $\displaystyle \lambda_2$, $\displaystyle \lambda_3$, which are equal.

    I know for a 2nd Order Recurrence Relation with equal roots, the general
    solution is: .$\displaystyle C_1$$\displaystyle \lambda^n$ $\displaystyle +$ $\displaystyle C_2$$\displaystyle n$$\displaystyle \lambda^n$, for some constants $\displaystyle C_1$, $\displaystyle C_2$.

    But what is the general solution for a 3rd Order with equal roots?

    It is: .$\displaystyle C_1$$\displaystyle \lambda^n$ + $\displaystyle C_2$$\displaystyle n$$\displaystyle \lambda^n$ $\displaystyle +$ $\displaystyle C_3$$\displaystyle n^2$$\displaystyle \lambda^n$

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