Initial rest condition for the linear constant-coefficient differential equations

1. What is linear differential equation with constant coefficients?
2. Do LTI systems have initial conditions?
3. Which response of LTI system does not depend on initial conditions?
4. How do you know if a differential equation is linear time invariant?

What is linear differential equation with constant coefficients?

A differential equation has constant coefficients if only constant functions appear as coefficients in the associated homogeneous equation. A solution of a differential equation is a function that satisfies the equation. The solutions of a homogeneous linear differential equation form a vector space.

Do LTI systems have initial conditions?

A causal LTI system has zero initial conditions and impulse response ℎ(𝑡). Its input (𝑡) and output (𝑡) are related through the linear constant-coefficient differential equation. d 2 y ( t ) d t 2 + α d y ( t ) d t + α 2 y ( t ) = x ( t ) .

Which response of LTI system does not depend on initial conditions?

Explanation: A LTI system is said to be memoryless only if it does not depend on any previous value of the input.

How do you know if a differential equation is linear time invariant?

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.