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Which of the following systems is NOT linear? ( )

Which of the following systems is NOT linear? ( )

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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**Question:** Which of the following systems is NOT linear? **Options:** (A) \( y(t) = (t^2 + 1) x(t) \) (B) \( y(t) = 2x(t) \) (C) \( \frac{dy}{dt} + y(t) + y^2(t) = \frac{dx}{dt} + x(t) \) (D) \( y(t) = x(t+2) \) **Explanation:** To determine which system is nonlinear, recall that a linear system must satisfy the principles of superposition and homogeneity. - Option (A) contains a time-dependent coefficient, \( t^2 + 1 \), multiplying the input \( x(t) \). This term doesn't violate linearity since \( (t^2 + 1) \) is simply a function of time, not \( y(t) \) or \( x(t) \). - Option (B) is linear as it involves a constant multiplication of the input \( x(t) \). - Option (C) includes \( y^2(t) \), which makes it nonlinear, as this violates the principle of superposition (output is not directly proportional to input). - Option (D) involves a delay operation (shifting \( x(t) \) by 2 units), which still maintains linearity. **Answer:** The system described in option (C) is NOT linear as it includes a squared term \( y^2(t) \).
Transcribed Image Text:**Question:** Which of the following systems is NOT linear? **Options:** (A) \( y(t) = (t^2 + 1) x(t) \) (B) \( y(t) = 2x(t) \) (C) \( \frac{dy}{dt} + y(t) + y^2(t) = \frac{dx}{dt} + x(t) \) (D) \( y(t) = x(t+2) \) **Explanation:** To determine which system is nonlinear, recall that a linear system must satisfy the principles of superposition and homogeneity. - Option (A) contains a time-dependent coefficient, \( t^2 + 1 \), multiplying the input \( x(t) \). This term doesn't violate linearity since \( (t^2 + 1) \) is simply a function of time, not \( y(t) \) or \( x(t) \). - Option (B) is linear as it involves a constant multiplication of the input \( x(t) \). - Option (C) includes \( y^2(t) \), which makes it nonlinear, as this violates the principle of superposition (output is not directly proportional to input). - Option (D) involves a delay operation (shifting \( x(t) \) by 2 units), which still maintains linearity. **Answer:** The system described in option (C) is NOT linear as it includes a squared term \( y^2(t) \).
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