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

630 Chapter 21 The Laplace transform
Solution From Table 21.1 wefind the results listedinTable 21.2.
Table21.2
f(t) F(s) f(t) F(s) f(t) F(s)
(a) t 3 6
s 4
(b) t 7 7!
s 8
(c) sin4t
4
s 2 +16
(d) e −2t 1
s +2
(e) cos(t/2)
(f) sinh3t
(g) cosh5t
s
s 2 +0.25
3
s 2 −9
s
s 2 −25
(h) tsin4t
(i) e −t sin2t
(j) e 3t cost
8s
(s 2 +16) 2
2
(s+1) 2 +4
s −3
(s−3) 2 +1
EXERCISES21.3
1 Determine the Laplacetransformsofthe following
functions:
(a) sin6t
( )
2t
(c) sin
3
(b) cos4t
( )
4t
(d) cos
3
(e) t 4 (f) t 2 t 3
(g) e −3t
(i)
1
e 4t
(h) e 3t
(j) tcos3t
(k) tsint
cos7t
(m)
e 5t
(o) cosh5t
(q) e −2t cosh7t
(s) e 7t cosh9t
(l) e −t sin3t
(n) sinh6t
(p) e −3t sinh4t
(r) e 4t sinh3t
2 ShowfromthedefinitionoftheLaplacetransformthat
L{u(t −d)} = e−sd ,d>0
s
Solutions
6
1 (a)
s 2 +36
6
(c)
9s 2 +4
24
(e)
s 5
1
(g)
s +3
(i)
1
s +4
(b)
(d)
(f)
(h)
(j)
s
s 2 (k)
+16
9s
9s 2 (m)
+16
120
s 6
(o)
1
s −3
(q)
s 2 −9
(s 2 +9) 2 (s)
2s
(s 2 +1) 2 (l)
s +5
(s+5) 2 +49
s
s 2 −25
s +2
(s+2) 2 −49
s −7
(s−7) 2 −81
(n)
(p)
(r)
3
(s+1) 2 +9
6
s 2 −36
4
(s+3) 2 −16
3
(s−4) 2 −9