Exponential Rate Problem: Solving for x and Showing Convergence to pi

[evolution685]

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

 

[Dick]

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.

 

[evolution685]

sweet. that saved me several hours of pounding my head.