584 Higher Engineering Mathematics
Table 61.1Elementary standard Laplace transforms
Function Laplace transforms
f(t) L{f(t)}=
∫∞
0 e
−stf(t)dt
(i) 1
1
s
(ii) k
k
s
(iii) eat
1
s−a
(iv) sinat
a
s^2 +a^2
(v) cosat
s
s^2 +a^2
(vi) t
1
s^2
(vii) t^2
2!
s^3
(viii) tn(n= 1 , 2 , 3 ,...)
n!
sn+^1
(ix) coshat
s
s^2 −a^2
(x) sinhat
a
s^2 −a^2
(a) L
{
1 + 2 t−
1
3
t^4
}
=L{ 1 }+ 2 L{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^2 t−3e−t}= 5 L(e^2 t)− 3 L{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 )
=
2 s+ 11
s^2 −s− 2
Problem 2. Find the Laplace transforms of:
(a) 6sin3t−4cos5t (b)2cosh2θ−sinh3θ.
(a) L{6sin3t−4cos5t}
= 6 L{sin3t}− 4 L{cos5t}
= 6
(
3
s^2 + 32
)
− 4
(
s
s^2 + 52
)
,
from(iv)and(v)ofTable61.1
=
18
s^2 + 9
−
4 s
s^2 + 25
(b) L{2cosh2θ−sinh3θ}
= 2 L{cosh 2θ}−L{sinh3θ}
= 2
(
s
s^2 − 22
)
−
(
3
s^2 − 32
)
from(ix)and(x)ofTable61.1
=
2 s
s^2 − 4
−
3
s^2 − 9
Problem 3. Prove that
(a)L{sinat}=
a
s^2 +a^2
(b)L{t^2 }=
2
s^3
(c)L{coshat}=
s
s^2 −a^2
(a) From equation (1),
L{sinat}=
∫∞
0
e−stsinatdt
=
[
e−st
s^2 +a^2
(−ssinat−acosat)
]∞
0
by integration by parts,
=
1
s^2 +a^2
[e−s(∞)(−ssina(∞)
−acosa(∞))−e^0 (−ssin0
−acos 0)]