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

640 Chapter 21 The Laplace transform
Havingwrittentheexpressionintermsofstandardforms,theinverseLaplacetransform
can now be found:
{ }
L −1 s −1
= 1 (
e −2t cos √ 1.5t − √ 3 e −2t sin √ )
1.5t
2s 2 +8s+11 2 1.5
In every case the given function ofsis written as a linear combination of standard
formscontained inTable 21.1.
EXERCISES21.6
1 Byusingthe standard formsin Table 21.1 findthe
inverseLaplacetransformsofthefollowingfunctions:
1 2
(a) (b)
s s 3
8 1
(c) (d)
s s +2
6 5
(e) (f)
s +4 s −3
7 2
(g) (h)
s +8 s 2 +4
s 6s
(i)
s 2 (j)
+9 s 2 +4
20 2
(k)
s 2 (l)
+16 (s +1) 2
8 9
(m)
(s +2) 2 (n)
(s +3) 3
e −3s 4e −6s
(o) (p)
s s
(q) e −9s (r) 4e −8s
4 s
(s)
s 2 (t)
−16 s 2 −9
12 6s
(u)
s 2 (v)
−9 s 2 −8
2
(w)
(s+1) 2 −4
2
(y)
(s+3) 2 −4
s +3
(x)
(s+3) 2 −4
6s
(z)
s 2 −5
2 Findthe inverse Laplacetransformsofthe following
functions:
3 4
(a) (b)
2s s − 1 s 3
30
1
(c)
s 2 (d)
3(s +2)
3s−7 s −6
(e)
s 2 (f)
+9 s −4
s +4 5
(g)
(s+4) 2 (h)
+1 (s+4) 2 +1
(i)
(k)
(m)
6s+17
(s+4) 2 +1
(j)
0.5
(s +0.5) 2 (l)
6s+9
s 2 +2s+10
(n)
s
s 2 +2s+7
s +5
s 2 +8s+20
7s+3
s 2 +4s+8
Solutions
1 (a) 1 (b) t 2
(o) u(t −3) (p) 4u(t −6)
(c) 8 (d) e −2t
(q) δ(t −9) (r) 4δ(t −8)
(e) 6e −4t (f) 5e 3t
(s) sinh4t (t) cosh3t
(g) 7e −8t (h) sin2t
(u) 4sinh3t (v) 6cosh √ 8t
(i) cos3t (j) 6cos2t
(w) e −t sinh2t (x) e −3t cosh2t
(k) 5sin4t (l) 2te −t
(y) e −3t sinh2t (z) 6cosh √ 5t
(m) 8te −2t 9
(n)
2 t2 e −3t