Time Invariant.
A system is time-invariant as long as the system shows certain fixed behaviors over time. For example, when x(t) shifts by a constant, y(t) should shift by the same constant.
$ y = x(t) $
$ x2 = x(t-t0) $
Then
$ y(t-t0) = x(t-t0) $
Also, the following should satisfy.
$ y = x(t) $
$ x2 = x(2t) $
Then
$ y(2t) = x(2t) $
Example of Time-Invariant System
$ y = sin(x) = sin(x(t)) $
When x(t) shifts by a constant t0,
$ x2(t) = x(t-t0) $
Then y(t) responds accordingly to the shift.
$ y2(t) = sin(x2) = sin(x(t-t0)) $
While maintaining, $ y2(t) = y(t-t0) $
Example of Time-Variant System
$ y = t * sin(x) = t * sin(x(t)) $
When x(t) shifts by a constant t0,
$ x2(t) = x(t-t0) $
Then y(t) does not respond accordingly to the shift.
$ y2(t) = t * sin(x2) = t * sin(x(t-t0)) $
Which is not equal to $ y2(t) = y(t-t0) = (t - t0) * sin(x(t-to)) $
