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Exponential rate problem

  1. Sep 14, 2008 #1
    first i had to show solve x′=sin(x) to get t=ln|(csc(x₀)+cot(x₀))/(csc(x)+cot(x))|

    i did that.

    next i need to show that for x₀=(π/4) you can solve x=2arctan(((e^{t})/(1+√2)))

    what i've done so far is

    t=ln|((csc(pi/4)+cot(pi/4))/(csc(x)+cot(x))|
    t=ln|((2/sqrt(2)+1)/(csc(x)+cot(x))|
    e^t=(2/sqrt(2)+1)/(csc(x)+cot(x))
    csc(x)+cot(x)=(2/sqrt(2)+1)/e^t

    and haven't been able to get any further. is this on the right track? how do i proceed?

    and finally i need to show that x(t)→pi as t→∞.

    the hint i've gotten is that it involves l'hopital's rule and maybe the equation

    lim t->inf (k/(1+(k/x0 - 1)e^(-mt)-k))/e^(-mt)

    i have no idea what this equation means or how it relates. anyone know?

    thanks a million
     
  2. jcsd
  3. Sep 14, 2008 #2

    Dick

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    To simplify there's a trig identity that says arccot(a/2)=csc(a)+cot(a). For the limit, just look at a graph of arctan.
     
  4. Sep 14, 2008 #3
    sweet. that saved me several hours of pounding my head.
     
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