### Problem StatementAn 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:\[ y(0^-) = 1, \quad \dot{y}(0^-) = 2 \]### ExplanationThis 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 \( t = 0^- \).

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.

Introductory Circuit Analysis (13th Edition)

13th Edition

ISBN:9780133923605

Author:Robert L. Boylestad

Publisher:Robert L. Boylestad

Chapter1: Introduction

Section: Chapter Questions

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Question

### 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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