A system is said to be linear if it follows the following conditions
1) The response to $ x_1(t) $ + $ x_2(t) $ is $ y_1(t) $ +$ y_2(t) $.
2) The response to $ ax_1(t) $ is $ ay_1(t) $, where a is any complex constant.
Example for a linear system is
y(t) = t x(t)
$ y_1 $ = 8$ e^t $
$ y_2 $=8$ t^2 $
Let ,
$ x_3 $ = 5$ e^t $ + 3$ t^2 $
The output is
$ p(t) $ = 8$ x_3 $
$ p(t) $=40 $ e^t $ + 24 $ t^2 $
$ p(t) $= 5$ x_1 $ + 3 $ x_2 $
The example for a nonlinear system is
y = $ x^2 (t) $
$ y_1 $ = $ t^2 $
$ y_2 $= $ sin^2 t $
Let ,
$ x_3 $ = $ t $ + $ sin t $
The output is
$ y(t) $ = $ x_3 $
$ y(t) $ = $ (t + sin t)^ 2 $
$ y(t) $= $ t^2 $ + $ sin^2 t $ + $ 2 t sin t $
