Problem 7. (B07) For the following systems, determine (i) if they are linear and (ii) if they are Time-Invariant (TI). Ti {x(t)} = y(t), s.t. D°y(t) + ty(t) = ¤(t) T2 {r(t)} = y(t), s.t. Dy(t – 1) + 3y(t) = ¤(t) T3 {r(t)} = y(t), s.t. D°y(t) – 2y²(t) = x(t) (За) (3b) (3c)
Problem 7. (B07) For the following systems, determine (i) if they are linear and (ii) if they are Time-Invariant (TI). Ti {x(t)} = y(t), s.t. D°y(t) + ty(t) = ¤(t) T2 {r(t)} = y(t), s.t. Dy(t – 1) + 3y(t) = ¤(t) T3 {r(t)} = y(t), s.t. D°y(t) – 2y²(t) = x(t) (За) (3b) (3c)
Introductory Circuit Analysis (13th Edition)
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Author:Robert L. Boylestad
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Chapter1: Introduction
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Transcribed Image Text:Problem 7. (B07) For the following systems, determine (i) if they are linear and (ii) if they are
Time-Invariant (TI).
Ti {x(t)} = y(t), s.t. D²y(t) + t²y(t) = x(t)
(За)
(3Ь)
T2 {x(t)} = y(t), s.t. Dy(t – 1) + 3y(t) = ¤(t)
T3 {x(t)} = y(t), s.t. D°y(t) – 2y²(t) = #(t)
T4 {x(t)} = y(t), s.t. Dy(t) + y(t) = x²(t)
T; {x(t)} = y(t), s.t. Dy(t) – 5y(t) = D¤(t) – 3x(t)
(Зс)
(3d)
(3e)
Hint: y(t) = L{x(t)} for this system means that the pair of signals (x, y) satisfies the given
Ordinary Differential Equation (ODE). To show that L is homogenous, you need to show that the
pair of amplitude-scaled signals (ax, ay) for any a e C satisfies the same ODE as the pair (x, y).
Use a similar rationale for the other two properties.
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