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)
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Question
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)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa145743f-9cae-4226-a3fd-bd208f7f38e5%2Fb6bd533e-a7e6-4b2f-9717-c98807bf72bd%2F4g5wkhb_processed.jpeg&w=3840&q=75)
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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