3. An LTI system is described by the following differential equation: d² d2z y(t) + 3 = y(t) + 2y(t) = x(t) + 2x(t) dt determine the total response of this system using Time-Domain Analysis given input x(t): =e-2tu(t) and initial conditions y(0) = 1, y(0) = 2.
3. An LTI system is described by the following differential equation: d² d2z y(t) + 3 = y(t) + 2y(t) = x(t) + 2x(t) dt determine the total response of this system using Time-Domain Analysis given input x(t): =e-2tu(t) and initial conditions y(0) = 1, y(0) = 2.
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![### Problem Statement
An LTI (Linear Time-Invariant) system is described by the following differential equation:
\[
\frac{d^2}{dt^2} y(t) + 3 \frac{d}{dt} y(t) + 2y(t) = \frac{d}{dt} x(t) + 2x(t)
\]
Determine the total response of this system **using Time-Domain Analysis** given the input:
\[
x(t) = e^{-2t}u(t)
\]
and the initial conditions:
\[
y(0^-) = 1, \quad \dot{y}(0^-) = 2
\]
### Explanation
This problem involves finding the total response of a Linear Time-Invariant (LTI) system using time-domain analysis, which incorporates both the natural response and the forced response of the system. The input to the system is a causal exponential function, \( e^{-2t}u(t) \), where \( u(t) \) is the unit step function. The initial conditions are specified for the system output \( y(t) \) and its first derivative at time \( t = 0^- \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F3f6c81c7-9a41-4014-8916-60df888d5014%2F41fdee45-0157-4b0b-9963-fa88b69e8f13%2F4mifzz_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Problem Statement
An LTI (Linear Time-Invariant) system is described by the following differential equation:
\[
\frac{d^2}{dt^2} y(t) + 3 \frac{d}{dt} y(t) + 2y(t) = \frac{d}{dt} x(t) + 2x(t)
\]
Determine the total response of this system **using Time-Domain Analysis** given the input:
\[
x(t) = e^{-2t}u(t)
\]
and the initial conditions:
\[
y(0^-) = 1, \quad \dot{y}(0^-) = 2
\]
### Explanation
This problem involves finding the total response of a Linear Time-Invariant (LTI) system using time-domain analysis, which incorporates both the natural response and the forced response of the system. The input to the system is a causal exponential function, \( e^{-2t}u(t) \), where \( u(t) \) is the unit step function. The initial conditions are specified for the system output \( y(t) \) and its first derivative at time \( t = 0^- \).
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