The Basics of Linearity
This example is solved using the following trigonometric identity:
$ cos(\omega t)=\frac{e^{\omega jt}+e^{-\omega jt}}{2}\, $.
We are told that the system has the following inputs and outputs:
$ x_1(t)=e^{2jt} \to y_1(t)=te^{2jt} $ , and
$ x_2(t)=e^{-2jt} \to y_2(t)=te^{-2jt} $
So what is the systems response to cos(2t)?
Using the identity:
$ x(t)=cos(2t)=\frac{e^{2jt}+e^{-2jt}}{2} \to y(t)=\frac{te^{2jt}+te^{-2jt}}{2}\, $
$ y(t)=t\frac{e^{2jt}+e^{-2jt}}{2}=tcos(2t)\, $
