first i had to show solve x′=sin(x) to get t=ln|(csc(x₀)+cot(x₀))/(csc(x)+cot(x))|(adsbygoogle = window.adsbygoogle || []).push({});

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

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# Exponential rate problem

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