Time Invariance
A system is time invariant if for a certain x(t) that produces an output y(t) if you shift the input to x(t-T) it just yields the same output shifted by the same T. y(t-T).
Time Invariant System
I propose that a system where
$ x(t) $ -> [SYSTEM] -> $ y(t) = 35x(t) $ is time invariant. Let's check.
Let $ x(t)=2e^t $ and $ T=5 $
$ x(t-T) = 2e^{t-5} $ -> [SYSTEM] -> $ 35*2*e^{t-5} $
$ x(t) = 2e^{t} $ -> [SYSTEM] -> $ y(t-T)=35*2*e^{t-5} $
As you can see these two outputs are the same, so the system is time invariant.
Time Variant System
I propose that a system where
$ x(t) $ -> [SYSTEM] -> $ x(35t) $
Is time variant. Let's use the same T and x(t) for this example.
$ x(t-T) = 2e^{t-5} $ -> [SYSTEM] -> $ 2*e^{35t-5} $
$ x(t) = 2e^{t} $ -> [SYSTEM] -> $ y(t-T)=2*e^{35(t-5)} = 2*e^{35t - 175} $
Clearly these two results are not the same, so the system is time variant.
