584 Higher Engineering Mathematics
Table 61.1 Elementary standard Laplace transforms
Function
Laplace transforms
f (t)
L{ f (t)}=
∞
0 e −st f (t) dt
(i) 1
1
s
(ii) k
k
s
(iii) e at
1
s − a
(iv) sin at
a
s 2 + a 2
(v) cos at
s
s 2 + a 2
(vi) t
1
s 2
(vii) t 2
2!
s 3
(viii) t n (n = 1, 2, 3, ...)
n!
s n+1
(ix) cosh at
s
s 2 − a 2
(x) sinh at
a
s 2 − a 2
(a) L
1 + 2t −
1
3
t
4
= L{1} + 2L{t } −
1
3
L{t
4
},
from equations (2) and (3)
=
1
s
+ 2
1
s 2
−
1
3
4!
s 4+1
,
from (i), (vi) and (viii) of Table 61.1
=
1
s
+
2
s 2 −
1
3
4 . 3 . 2 . 1
s 5
=
1
s
+
2
s 2 −
8
s 5
(b) L{5e 2t − 3e −t } = 5L(e 2t ) − 3L{e −t },
from equations (2) and (3)
= 5
1
s − 2
− 3
1
s − (−1)
,
from (iii) of Table 61.1
=
5
s − 2
−
3
s + 1
=
5(s + 1) − 3(s − 2)
(s − 2)(s + 1)
=
2s + 11
s 2 − s − 2
Problem 2. Find the Laplace transforms of:
(a) 6 sin 3t − 4 cos5t (b) 2 cosh 2θ − sinh 3θ.
(a) L{6 sin3t − 4 cos5t }
= 6L{sin 3t } − 4L{cos5t }
= 6
3
s 2 + 3 2
− 4
s
s 2 + 5 2
,
from (iv) and (v) of Table 61.1
=
18
s 2 + 9
−
4s
s 2 + 25
(b) L{2 cosh 2θ − sinh 3θ}
= 2L{cosh 2θ} − L{sinh 3θ}
= 2
s
s 2 − 2 2
−
3
s 2 − 3 2
from (ix) and (x) of Table 61.1
=
2s
s 2 − 4
−
3
s 2 − 9
Problem 3. Prove that
(a) L{sin at } =
a
s 2 + a 2 (b) L{t
2
} =
2
s 3
(c) L{cosh at } =
s
s 2 − a 2
(a) From equation (1),
L{sin at } =
∞
0
e
−st sin at dt
=
e −st
s 2 + a 2 (−s sin at − a cos at )
∞
0
by integration by parts,
=
1
s 2 + a 2 [e
−s(∞)
(−s sin a(∞)
− a cos a(∞)) − e
0
(−s sin 0
−a cos 0)]