Time-invariant differential equation

1. What is a time invariant differential equation?
2. How do you know if an equation is time invariant?
3. What is meant by time invariant?
4. Is a derivative time invariant?

What is a time invariant differential equation?

A linear differential equation with constant coefficients displays time invariance. If we use the same input and starting conditions for a system now or at some later time then the result relative to the initial starting time will be identical.

How do you know if an equation is time invariant?

A system is time-invariant if its output signal does not depend on the absolute time. In other words, if for some input signal x(t) the output signal is y1(t)=Trx(t), then a time-shift of the input signal creates a time-shift on the output signal, i.e. y2(t)=Trx(t−t0)=y1(t−t0).

What is meant by time invariant?

Mathematically speaking, "time-invariance" of a system is the following property: Given a system with a time-dependent output function , and a time-dependent input function , the system will be considered time-invariant if a time-delay on the input directly equates to a time-delay of the output function.

Is a derivative time invariant?

The time-derivative operator from calculus and the act of integration over time are both linear, time-invariant processes. A time-derivative is just a running difference between two values slightly separated in time, then scaled by 1/Δt.