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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)
Introductory Circuit Analysis (13th Edition)
13th Edition
ISBN:9780133923605
Author:Robert L. Boylestad
Publisher:Robert L. Boylestad
Chapter1: Introduction
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### 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}
Transcribed Image Text:### 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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