082-Engineering-Mathematics-Anthony-Croft-Robert-Davison-Martin-Hargreaves-James-Flint-Edisi-5-2017

678 Chapter 21 The Laplace transform
Solution L{f (t)} =
=
∫ 1.5
e −st f (t)dt
0
1 −e −1.5s
∫ 1.25
2e −st dt + ∫ 1.5
0 1.25 0e−st dt
1 −e −1.5s
= 2(1 −e−1.25s )
s(1 −e −1.5s )
EXERCISES21.13
1 Find the Laplacetransformsof
(a) 3u(t) + δ(t) (b) −6u(t) +4δ(t)
(c) 3u(t −2)+δ(t −2)
(d) u(t −3)−δ(t −4)
1
(e) 2 u(t −4)+3δ(t −4)
whereu(t)is the unitstepfunction.
2 Find the inverse Laplace transformsof
(a) 2 s −1 (b) 2
3s + 1 2
(d) 4s−3
s
3 Aperiodicwaveform is defined by
{
t 0t1
f(t)=
2−t 1<t2
(c) 3−2s
s
andhas aperiod of2.
(a) Sketch two cycles of f (t).
(b) Find L{f (t)}.
4 Awaveform, f (t),is defined by
{
2 0t1.5
f(t)=
−2t+5 1.5<t<2.5
andhas aperiod of2.5.
(a) Sketch f (t)on [0,5].
(b) Find L{f (t)}.
5 Ifthe impulseresponse ofanetwork isg(t) = 10e −4t
findthe outputwhenthe input is f (t) = e −t cos2t,
t0.
Solutions
3
1 (a)
s +1 (b) −6 s +4
3e −2s
(c) +e −2s (d) e−3s −e −4s
s s
e −4s
(e) +3e −4s
2s
2 (a)2u(t)−δ(t) (b) 2u(t) + δ(t)
3 2
3 (b)
4 (b)
5
(c)3u(t) −2δ(t)
1−e −s
s 2 (1+e −s )
2(s +e −2.5s −e −1.5s )
s 2 (1 −e −2.5s )
(30cos2t +20sin2t)e −t
13
(d) 4δ(t) −3u(t)
− 30e−4t
13
REVIEWEXERCISES21
1 GivenF(s) = s +1
s 2 isthe Laplace transformof
+2
f (t),findthe Laplace transformsofthe following:
(a) f(t)
e 2t
(b) 3e 2t f(t)
(c)2e −t (f(t)+1) (d) u(t−1)f(t−1)
(e)4u(t −3)f(t −3)
(f) e −2t u(t −2)f(t −2)