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Math Help - direct linearisation

  1. #1
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    direct linearisation

    Use the method of direct linearisation to determine the stability of
    (a) the fixed point at x = pi for the differential equation dx/dt = sin x, and
    (b) the only fixed point for the differential equation dx/dt = ln (x/a) , a > 0.
    (Hint: ln(1 + z) ~ z for |z| < 1.)
    In each case verify your answer using the method of linearisation by Taylor polynomial.
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  2. #2
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    Quote Originally Posted by mamt6 View Post
    Use the method of direct linearisation to determine the stability of
    (a) the fixed point at x = pi for the differential equation dx/dt = sin x,
    Two things you learned long ago are that \lim_{x\to 0}\frac{sin x}{x}= 1 and that sin(x-\pi)= -sin(x) so \lim_{x\to \pi}\frac{sin (x-\pi)}{x-\pi}= \lim_{x\to 0}\frac{-sin(x)}{x}= -1 so that sin x is approximately -(x-\pi) for x close to \pi.

    and
    (b) the only fixed point for the differential equation dx/dt = ln (x/a) , a > 0.
    (Hint: ln(1 + z) ~ z for |z| < 1.)
    In each case verify your answer using the method of linearisation by Taylor polynomial.
    x/a= 1+ (x/a- 1) so ln(x/a) is approximately x/a- 1. That's the linearization you want.
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