Definition of Time Invariance
A time-invariant system is one whose output does not depend explicitly on time. If the input signal x produces an output y then any time shifted input results in a time-shifted output.
Examples
Time-Invariant System:
$ y(t) = x(t) + 5 $
$ x(t) \to system \to timedelay \to y(t) = x(t - t_{0}) + 5 $
$ x(t) \to timedelay \to system \to y(t) = x(t - t_{0}) + 5 $
Non-Time-Invariant System:
$ y(t) = x(5t) + 5 $
$ x(t) \to system \to timedelay \to y(t) = x(5t - t_{0}) + 5 $
$ x(t) \to timedelay \to system \to y(t) = x(5(t - t_{0})) + 5 $
