Introduction to Laplace transforms 585
=
1
s^2 +a^2
[( 0 )− 1 ( 0 −a)]
=
a
s^2 +a^2
(provideds> 0 )
(b) From equation (1),
L{t^2 }=
∫∞
0
e−stt^2 dt
=
[
t^2 e−st
−s
−
2 te−st
s^2
−
2e−st
s^3
]∞
0
by integration by parts twice,
=
[
( 0 − 0 − 0 )−
(
0 − 0 −
2
s^3
)]
=
2
s^3
(provideds> 0 )
(c) From equation (1),
L{coshat}=L
{
1
2
(eat+e−at)
}
,
from Chapter 5
=
1
2
L{eat}+
1
2
L{e−at},
equations (2) and (3)
=
1
2
(
1
s−a
)
+
1
2
(
1
s−(−a)
)
from (iii) of Table 61.1
=
1
2
[
1
s−a
+
1
s+a
]
=
1
2
[
(s+a)+(s−a)
(s−a)(s+a)
]
=
s
s^2 −a^2
(provideds>a)
Problem 4. Determine the Laplace transforms of:
(a) sin^2 t (b) cosh^23 x.
(a) Since cos2t= 1 −2sin^2 tthen
sin^2 t=
1
2
( 1 −cos2t). Hence,
L{sin^2 t}=L
{
1
2
( 1 −cos2t)
}
=
1
2
L{ 1 }−
1
2
L{cos2t}
=
1
2
(
1
s
)
−
1
2
(
s
s^2 + 22
)
from (i) and (v) of Table 61.1
=
(s^2 + 4 )−s^2
2 s(s^2 + 4 )
=
4
2 s(s^2 + 4 )
=
2
s(s^2 +4)
(b) Since cosh 2x=2cosh^2 x−1then
cosh^2 x=
1
2
( 1 +cosh2x)from Chapter 5.
Hence cosh^23 x=
1
2
( 1 +cosh 6x)
ThusL{cosh^23 x}=L
{
1
2
( 1 +cosh 6x)
}
=
1
2
L{ 1 }+
1
2
L{cosh6x}
=
1
2
(
1
s
)
+
1
2
(
s
s^2 − 62
)
=
2 s^2 − 36
2 s(s^2 − 36 )
=
s^2 − 18
s(s^2 −36)
Problem 5. Find the Laplace transform of
3sin(ωt+α),whereωandαare constants.
Using the compound angle formula for sin(A+B),
from Chapter 17, sin(ωt+α)may be expanded to
(sinωtcosα+cosωtsinα). Hence,
L{3sin(ωt+α)}
=L{ 3 (sinωtcosα+cosωtsinα)}
=3cosαL{sinωt}+3sinαL{cosωt},
sinceαis a constant
=3cosα
(
ω
s^2 +ω^2
)
+3sinα
(
s
s^2 +ω^2
)
from (iv) and (v) of Table 61.1
=
3
(s^2 +ω^2 )
(ωcosα+ssinα)
Now try the following exercise
Exercise 219 Further problems on an
introduction to Laplace transforms
Determine the Laplace transforms in Problems
1to9.