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voltage across the inductor is L (dyo/dt) or yo(t), this equation can be written as follows: + 3yo(t) + v ert) = 0 yo(O) + 3yo(0) + ve(O) Setting t = 0, we o btain
=0 (2.14) But yo(O) = 0 a nd vc(O) = 5. Consequently,
Yo(O) = -5 Therefore, the desired initial conditions are
Yo(O) = 0 and Yo(O) = -5 Thus, the problem reduces to finding yo(t), the zero-input component of y(t) o fthe system
specified by the equation (D2 + 3 D + 2)y(t) = D f(t), when the initial conditions are
yo(O) = 0 and yo(O) = - 5. We have already solved this problem in Example 2.1a, where
yo(t) = - 5e- t + 5 e- 2t
This is the zero-input component of the loop current y (t).
I t will be interesting to find the initial conditions a t t = 0 - and 0+ for the total
response y (t). Let us compare y(O-) and y(O-) with y(O+) and Y(O+). The two pairs can
be compared by writing the loop equation for the circuit in Fig. 2.1a a t t = 0 - and t = 0+.
The only difference between the two situations is t hat a t t = 0 -, the input f (t) = 0,
whereas a t t = 0 +, the input f (t) = 10 (because f (t) = l Oe- 3t ). Hence, the two loop
equations are + 3 y(0-) + v e(O-) =
y(O+) + 3y(0+) + vc(O+) =
y (O-) 0
10 The loop current y(O+) = y(O-) = 0 because it cannot change instantaneously in the
absence of impulsive voltage. The same is true of the capacitor voltage. Hence, ve(O+) =
v c(O-) = 5. S ubstituting these values in the above equations, we obtain y (O-) = - 5 and
y(O+) = 5. Thus
2, E xercise E 2.3 • In the circuit in Fig. 2.1a, the inductance L = 0 and the initial capacitor voltage vc(O) = 30
volts. Show that the zero-input component of the loop current is given by yo(t) = _10e- 2t / 3 for
t ~ O. V' 113 I n t his example we c omputed t he z ero-input component without using t he i nput
f (t). T he z ero-state component c an b e c omputed from t he knowledge of t he i nput
f (t) alone; t he i nitial conditions are assumed t o b e zero (system in zero s tate). T he
two components of t he s ystem response (the zero-input a nd z ero-state components)
are independent of each other. The two worlds o f z ero-input response a nd z ero-state
response coexist s ide by side, n either o f t hem knowing o r c aring what the other is
doing. For each component, the other is t otally irrelevant.
Role of Auxiliary Conditions in Solution o f Differential Equations F ig. 2 .1 yo(t) S ystem Response t o I nternal Conditions: Zero-Input Response Independence of Zero-Input and Z ero-State Response 30 IH 2.2 S olution of a differential equation requires additional pieces of information (the
a uxiliary c onditions). W hy? We now show t hat a differential equation does not,
i n general, have a unique solution unless some additional constraints (or conditions)
o n t he solution a re known. T he r eason is t hat, as explained in t he discussion on
invertibility (Sec. 1.7), t he d ifferentiation o peration is n ot invertible unless one piece
of information a bout y (t) is given. Thus, differentiation is a n irreversible (noninvertible) o p...
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