Page 96 - Schaum's Outline of Theory and Problems of Signals and Systems

CHAP. 21 LINEAR TIME-INVARIANT SYSTEMS 85

2.20. Consider a continuous-time system whose input x(t) and output y(t) are related by

where a is a constant.

(a) Find y(t) with the auxiliary condition y(0) = y, and
x(t) = ~e-~'u(t) (2.102)
(b) Express y(t) in terms of the zero-input and zero-state responses.

(a) Let

where yp(t) is the particular solution satisfying Eq. (2.101) and yh(t) is the homogeneous
solution which satisfies

Assume that
yp(t) =~e-~' t>O
Substituting Eq. (2.104) into Eq. (2.101), we obtain
- b~e-6' + a~e-b' = K~-~'

from which we obtain A = K/(a - b), and

To obtain yh(t), we assume
yh(t) = BeS'

Substituting this into Eq. (2.103) gives
sBe"+aBe"=(s+a)Be"'=O
from which we have s = -a and

yh(t) = Be-"'
Combining yp(t) and yh(t), we get
K
y(t) =Be-"+ - t>O
e-b~
a-b
From Eq. (2.106) and the auxiliary condition y(O) =yo, we obtain
K
B=yo- -
a-b
Thus, Eq. (2.106) becomes