Page 163 - Schaum's Outline of Theory and Problems of Signals and Systems

LAPLACE TRANSFORM AND CONTINUOUS-TIME LTI SYSTEMS [CHAP. 3

(b) Show that

(a) Let x(t) have unilateral Laplace transform X,?(s). Using Eq. (3.99) and integrating by
parts, we obtain

Thus, we have

(b) By definition (3.99)
Oe OC
P+{u(t)) = ,/ u(t)e-"dt= ,/ e-"dt
0' 0'
1
Re(s) > 0

From Eq. (1.30) we have

Taking the 0' unilateral Laplace transform of Eq. (3.103) and using Eq. (3.100), we
obtain

This is consistent with Eq. (1.21); that is,

Note that taking the 0 unilateral Laplace transform of Eq. (3.103) and using Eq. (3.44),
we obtain

APPLICATION OF UNILATERAL LAPLACE TRANSFORM

337. Using the unilateral Laplace transform, redo Prob. 2.20.
The system is described by
~'(t) +ay(t) =x(l)
1.
t
with y(0) = yo and x(t = ~e-~'u(