Definition
A time invariant system is a system whose response to the shifted input $ x(t-t_0) $ is the shifted output $ y(t-t_0) $ for any input x(t) and any time $ t_0 \varepsilon \real $
Time Invariant Example
First take the equation: $ y(t)=17x(t) $
Consider an input: $ y_1(t)=17x_1(t) $
Now, shift that input in time: $ x_2 (t)= x_1(t-t_0) $
$ y_2(t)=17x_2(t)= 17x_1(t-t_0) $
$ y_1(t-t_0)=17x_1(t-t_0) $
Note that $ y_2(t) = y_1(t-t_0) $, this means that the system is time invariant
Time Variant Example
First take the equation: $ y(t)=17tx(t) $
Consider an input: $ y_1(t)=17tx_1(t) $
Now, shift that input in time: $ x_2 (t)= x_1(t-t_0) $
$ y_2(t)=17tx_2(t)= 17tx_1(t-t_0) $
$ y_1(t-t_0)=17(t-t_0)x_1(t-t_0) $
Note that $ y_2(t) \ne y_1(t-t_0) $, this means that the system is time variant
