Problem
Given that the response to $ x(t)=exp(2jt) $ is $ y(t)=t*exp(-2jt) $, and its response to $ x(t)=exp(-2jt) $ is $ y(t)=t*exp(2jt) $. What is the systems response to $ x(t)=cos(2t) $?
Solution
If a system is linear we know that:
$ ax(t)+bx(t)=ay(t)+by(t) $
Using Euler's Method we know that:
$ exp(jt)=cos(t)+jsin(t) $
Given this:
$ exp(2jt)+exp(-2jt)=cos(2t)+j*sin(2t)+cos(-2t)+j*sin(-2t) $
$ exp(2jt)+exp(-2jt)=cos(2t)+j*sin(2t)+cos(2t)-j*sin(2t) $
$ exp(2jt)+exp(-2jt)=2cos(2t) $
So now:
$ .5*2cos(2t)=cos(2t) $
$ cos(2t)-->.5*(t*exp(2jt)+t*exp(-2jt)) $
$ cos(2t)--> t*cos(2t) $
