TIME INVARIANCE
Time invariance, in my definition, is such a system that does not stretch or shrink the input function and does not change time shift of input is called "time invariance."
Example of Time invariant system and its proof
$ \,y(t)=e^{x(t)}\, $
Proof:
$ x(t) \to System \to y(t)=e^{x(t)} \to Time Shift(t0) \to z(t)=y(t-t0) $
$ \, =e^{x(t-t0)}\, $
$ x(t) \to Time Shift(t0) \to y(t)=x(t-t0) \to System \to z(t)=e^{y(t)} $
$ \, =e^{x(t-t0)}\, $
Both cascades yielded the same outputs, thus $ \,y(t)=e^{x(t)}\, $ is time invariant.
Example of Time variant system and its proof
$ \,y(t)=x(2t)\, $
Proof:
$ x(t) \to System \to y(t)=x(2t) \to Time Shift(t0) \to z(t)=y(t-t0) $
$ \, =x(2t-2t0)\, $
$ x(t) \to Time Shift(t0) \to y(t)=x(t-t0) \to System \to z(t)=y(2t) $
$ \, =x(2t-t0)\, $
They yielded different outputs, thus $ \,y(t)=x(2t)\, $ is time-variant.
