Definition
A time invariant system is simply a system whose output does not depend on time.
Said mathematically, a system with input $ x(t) $ and output $ y(t) $, such that any time-shifted input $ x(t+T) $ results in an output y(t+T).
Examples
$ y(t)=5x(t) $
The above system is time invariant because the output, $ y $, doesn't depend on $ t $ explicitly. One easy way to test if a system is time invariant is the take the derivative, in this case it would be a constant, 5, so the system is invariant of time.
$ y(t)=t^2x(t) $
Conversely, this system is time variant. This is easily seen because as the time $ t $ increases, $ y(t) $ increases with $ t^2 $. As in the above example, check by taking the derivative and here we get $ 2t $ which is not a constant so the system varies with time.
