Phukraut, your system that you posted looks promising. But
f<sub>1</sub>(x) = 1 by the way I am looking at w<sup>(n)</sup>(x).
w<sup>(n)</sup>(x) = <sup>d<sup>n-1</sup></sup>/<sub>dx<sup>n-1</sup></sub> e<sup>sin(x)</sup>
So e^sin(x) is th first deriviative.