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the zero-state component doub EXAMPLE 1.9 Show that the system described by the equation dy +3y(1) = x(t) dt (1.43) is linear It] Let the system response to the inputs x1(t) and x2(t) be y, (t) and y2(t), respectively. Then dy 3u.(t)

the zero-state component doub EXAMPLE 1.9 Show that the system described by the equation dy +3y(1) = x(t) dt (1.43) is linear It] Let the system response to the inputs x1(t) and x2(t) be y, (t) and y2(t), respectively. Then dy 3u.(t)

Linear Algebra: A Modern Introduction
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter4: Eigenvalues And Eigenvectors
Section4.6: Applications And The Perron-frobenius Theorem
Problem 70EQ
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Explane for me by steps please because i did not understand
y(t). decomposition property, linearity implies that both the zero-input and zero-state components must obey the of superposition with respect to each of their respective causes. For example, if we increase the initial condition k-fold, the ze component must also increase k-fold. Similarly, if we increase the input k-fold, the zero-state component must also increase These facts can be readily verified from Eq. (1.42) for the RC circuit in Fig. 1.26. For instance, if we double the initial conditic the zero-input component doubles; if we double the input x(t), the zero-state component doubles. EXAMPLE 1.9 Show that the system described by the equation dy + 3y(t) = x(t) dt (1.43) is linear.(t) Let the system response to the inputs x1(t) and x2(t) be y1(t) and y2(t), respectively. Then dy + 3yı (1) = x (1) dt and dy +3y2(f) = x2(1) dt %3D Multiplying the first equation by k1, the second with k2, and adding them ylelds Ik yı (1) +ky2())+3[k y(1) + ky2(1)] = k,x(1) + kzx2(t) But this equation is the system equation [Eq. (1.43)] with x(t) = kx1(1) + kzx2(1) and y(t) = kjyı (t) + k2y2(1)
Transcribed Image Text:y(t). decomposition property, linearity implies that both the zero-input and zero-state components must obey the of superposition with respect to each of their respective causes. For example, if we increase the initial condition k-fold, the ze component must also increase k-fold. Similarly, if we increase the input k-fold, the zero-state component must also increase These facts can be readily verified from Eq. (1.42) for the RC circuit in Fig. 1.26. For instance, if we double the initial conditic the zero-input component doubles; if we double the input x(t), the zero-state component doubles. EXAMPLE 1.9 Show that the system described by the equation dy + 3y(t) = x(t) dt (1.43) is linear.(t) Let the system response to the inputs x1(t) and x2(t) be y1(t) and y2(t), respectively. Then dy + 3yı (1) = x (1) dt and dy +3y2(f) = x2(1) dt %3D Multiplying the first equation by k1, the second with k2, and adding them ylelds Ik yı (1) +ky2())+3[k y(1) + ky2(1)] = k,x(1) + kzx2(t) But this equation is the system equation [Eq. (1.43)] with x(t) = kx1(1) + kzx2(1) and y(t) = kjyı (t) + k2y2(1)
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