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 theof superposition with respect to each of their respective causes. For example, if we increase the initial condition k-fold, the zecomponent must also increase k-fold. Similarly, if we increase the input k-fold, the zero-state component must also increaseThese facts can be readily verified from Eq. (1.42) for the RC circuit in Fig. 1.26. For instance, if we double the initial conditicthe zero-input component doubles; if we double the input x(t), the zero-state component doubles.EXAMPLE 1.9Show that the system described by the equationdy+ 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. Thendy+ 3yı (1) = x (1)dtanddy+3y2(f) = x2(1)dt%3DMultiplying the first equation by k1, the second with k2, and adding them yleldsIk yı (1) +ky2())+3[k y(1) + ky2(1)] = k,x(1) + kzx2(t)But this equation is the system equation [Eq. (1.43)] withx(t) = kx1(1) + kzx2(1)andy(t) = kjyı (t) + k2y2(1)

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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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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)

Expert Solution

Step 1

System of linear equations:

A set of two or more linear equations is referred to as a system of linear equations.
The graph of a system of two equations in two variables (x and y) is a pair of lines in the plane.
There are three alternatives:
At zero locations, the lines cross.
The lines come together at a single location. (In the vast majority of cases.)
The lines cross at an unlimited number of spots. (The lines in the two equations are the same.)