### Analysis of Linear Systems Described by Differential Equations#### Problem Statement:For the systems described by the following equations, with the input \( x(t) \) and output \( y(t) \), determine whether the systems are linear, causal, time-invariant, and dynamic. Also, find the orders of the systems.a) \[ \frac{d^3y}{dt^3} + y(t) = x(t + 2) \]b)\[ \left( \frac{dy}{dt} \right)^3 + \frac{d^2y}{dt^2} + y(t) = x(t - 2) \]c) (crossed out)\[ \frac{d^5y}{dt^5} - \frac{d^3y}{dt^3} + y(t) = x(t + 2) \]d) (crossed out)\[ y^2(t) + \sin(y(t)) = x^2(t) \]#### Property Analysis:Below is a table summarizing the properties and orders of the systems:| Property | Systems ||--------------------|---------|| | a | b | c | d || Linear (Y/N) | | | | || Causal (Y/N) | | | | || Time-Invariant (Y/N) | | | | || Dynamic (Y/N) | | | | || Order | | | | |#### Explanation:- **Linear System**: A system is linear if it obeys superposition and scaling principles.- **Causal System**: A system is causal if its output at any time depends only on values of the input at the present time and in the past, not the future.- **Time-Invariant System**: A system is time-invariant if its behavior and characteristics do not change over time.- **Dynamic System**: A system is dynamic if its output depends on past or future inputs, as opposed to just the present input.- **Order of the System**: The order is determined by the highest derivative of \( y(t) \) in the differential equation.#### Analyzing Each System:- **System a)**: \[ \frac{d^3y}{dt^3}

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4. For the systems described by the following equations, with the input x(t) and output y(t), determine whether the systems are linear, causal, time-invariant and dynamic. Also, find the orders of systems. d³y a) dt³ + y(t) = x(t+2) 3 dy d² y dt² b) dt -+ y(t) = x(t-2)

4. For the systems described by the following equations, with the input x(t) and output y(t), determine whether the systems are linear, causal, time-invariant and dynamic. Also, find the orders of systems. d³y a) dt³ + y(t) = x(t+2) 3 dy d² y dt² b) dt -+ y(t) = x(t-2)

Introductory Circuit Analysis (13th Edition)

13th Edition

ISBN:9780133923605

Author:Robert L. Boylestad

Publisher:Robert L. Boylestad

Chapter1: Introduction

Section: Chapter Questions

Problem 1P: Visit your local library (at school or home) and describe the extent to which it provides literature...

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