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Math Help - Solve Linear Difference equation

  1. #1
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    Solve Linear Difference equation

    Dear all, the following ar Berkely problem (Spring 85).

    Let be given. Consider the linear difference equation





    (Note the analogy with the differential equation .)
    1. Find the general solution of by trying suitable exponential substitutions.
    2. Find the solution with and . Denote it by ,
      .
    3. Let be fixed and . Show that

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  2. #2
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    Re: Solve Linear Difference equation

    The first one I got the solution:

    y(nh)=(1+h^2)^\frac{n}{2}[C_1 \cos n\phi+C_2 \sin n\phi]

    Here,
    \phi=\arctan h.
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  3. #3
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    Re: Solve Linear Difference equation

    2. I find that
    S_h(nh)=(1+h^2)^\frac{n-1}{2}\frac{\sin n\phi}{\sin\phi}.

    3. I then confused then. Since I could only verify that
    \lim_{n\to\infty}\sin [n\arctan \frac{x}{n}]=\sin x,
    while
    \lim_{n\to\infty}S_{x/n}(nx/n)=\lim_{n\to\infty}(1+x^2/n^2)^{(n-1)/2}\frac{\sin [n\arctan \frac{x}{n}]}{\sin [\arctan\frac{x}{n}]}=?

    In my calculations, it seems to be \infty. What's going wrong?

    Would you help me? Thank you.
    Last edited by Ackbeet; August 6th 2011 at 02:55 AM.
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