Properties of Laplace transforms 589
=
3
2 s+ 1
−
6 s+ 3
4 s^2 + 4 s+ 17
=
3 ( 4 s^2 + 4 s+ 17 )−( 6 s+ 3 )( 2 s+ 1 )
( 2 s+ 1 )( 4 s^2 + 4 s+ 17 )
=
12 s^2 + 12 s+ 51 − 12 s^2 − 6 s− 6 s− 3
( 2 s+ 1 )( 4 s^2 + 4 s+ 17 )
=
48
(2s+1)(4s^2 + 4 s+17)
Now try the following exercise
Exercise 220 Further problemson Laplace
transforms of the formeatf(t)
Determine theLaplace transforms of the following
functions:
- (a) 2te^2 t(b)t^2 et
[
(a)
2
(s− 2 )^2
(b)
2
(s− 1 )^3
]
- (a) 4t^3 e−^2 t(b)
1
2
t^4 e−^3 t
[
(a)
24
(s+ 2 )^4
(b)
12
(s+ 3 )^5
]
- (a) etcost(b) 3e^2 tsin2t
[
(a)
s− 1
s^2 − 2 s+ 2
(b)
6
s^2 − 4 s+ 8
]
- (a) 5e−^2 tcos3t(b) 4e−^5 tsint
[
(a)
5 (s+ 2 )
s^2 + 4 s+ 13
(b)
4
s^2 + 10 s+ 26
]
- (a) 2etsin^2 t(b)
1
2
e^3 tcos^2 t
⎡
⎢
⎢
⎣
(a)
1
s− 1
−
s− 1
s^2 − 2 s+ 5
(b)
1
4
(
1
s− 3
+
s− 3
s^2 − 6 s+ 13
)
⎤
⎥
⎥
⎦
- (a) etsinht(b) 3e^2 tcosh4t
[
(a)
1
s(s− 2 )
(b)
3 (s− 2 )
s^2 − 4 s− 12
]
- (a) 2e−tsinh3t(b)
1
4
e−^3 tcosh 2t
[
(a)
6
s^2 + 2 s− 8
(b)
s+ 3
4 (s^2 + 6 s+ 5 )
]
- (a) 2et(cos 3t−3sin3t)
(b) 3e−^2 t(sinh2t−2cosh2t)
[
(a)
2 (s− 10 )
s^2 − 2 s+ 10
(b)
− 6 (s+ 1 )
s(s+ 4 )
]
62.3 The Laplace transforms of
derivatives
(a) First derivative
Let the first derivative of f(t)be f′(t)then, from
equation (1),
L{f′(t)}=
∫∞
0
e−stf′(t)dt
From Chapter 43, when integrating by parts
∫
u
dv
dt
dt=uv−
∫
v
du
dt
dt
When evaluating
∫∞
0 e
−stf′(t)dt,
letu=e−stand
dv
dt
=f′(t)
from which,
du
dt
=−se−standv=
∫
f′(t)dt=f(t)
Hence
∫∞
0
e−stf′(t)dt
=
[
e−stf(t)
]∞
0 −
∫∞
0
f(t)(−se−st)dt
=[0−f( 0 )]+s
∫∞
0
e−stf(t)dt
=−f( 0 )+sL{f(t)}
assuming e−stf(t)→0ast→∞,andf( 0 )isthevalue
off(t)att=0. Hence,
L{f′(t)}=sL{f(t)}−f(0)
or L
{
dy
dx
}
=sL{y}−y(0)
⎫
⎬
⎭
(3)
wherey( 0 )is the value ofyatx=0.