Inverse Laplace transforms 595
(a)L−^1
{
3
s^2 − 4 s+ 13
}
=L−^1
{
3
(s− 2 )^2 + 32
}
=e^2 tsin3t,
from (xii) of Table 63.1
(b)L−^1
{
2 (s+ 1 )
s^2 + 2 s+ 10
}
=L−^1
{
2 (s+ 1 )
(s+ 1 )^2 + 32
}
=2e−tcos3t,
from (xiii) of Table 63.1
Problem 6. Determine
(a)L−^1
{
5
s^2 + 2 s− 3
}
(b)L−^1
{
4 s− 3
s^2 − 4 s− 5
}
(a)L−^1
{
5
s^2 + 2 s− 3
}
=L−^1
{
5
(s+ 1 )^2 − 22
}
=L−^1
⎧
⎪⎨
⎪⎩
5
2
( 2 )
(s+ 1 )^2 − 22
⎫
⎪⎬
⎪⎭
=
5
2
e−tsinh2t,
from (xiv) of Table 63.1
(b) L−^1
{
4 s− 3
s^2 − 4 s− 5
}
=L−^1
{
4 s− 3
(s− 2 )^2 − 32
}
=L−^1
{
4 (s− 2 )+ 5
(s− 2 )^2 − 32
}
=L−^1
{
4 (s− 2 )
(s− 2 )^2 − 32
}
+L−^1
{
5
(s− 2 )^2 − 32
}
=4e^2 tcosh3t+L−^1
⎧
⎪⎨
⎪⎩
5
3
( 3 )
(s− 2 )^2 − 32
⎫
⎪⎬
⎪⎭
from (xv) of Table 63.1
=4e^2 tcosh 3t+
5
3
e^2 tsinh3t,
from (xiv) of Table 63.1
Now try the following exercise
Exercise 223 Further problemson inverse
Laplace transformsof simple functions
Determine the inverse Laplace transforms of the
following:
- (a)
7
s
(b)
2
s− 5
[(a) 7 (b) 2e^5 t]
- (a)
3
2 s+ 1
(b)
2 s
s^2 + 4
[
(a)
3
2
e−
1
2 t (b)2cos2t
]
- (a)
1
s^2 + 25
(b)
4
s^2 + 9
[
(a)
1
5
sin5t (b)
4
3
sin3t
]
- (a)
5 s
2 s^2 + 18
(b)
6
s^2
[
(a)
5
2
cos3t (b) 6 t
]
- (a)
5
s^3
(b)
8
s^4
[
(a)
5
2
t^2 (b)
4
3
t^3
]
- (a)
3 s
1
2
s^2 − 8
(b)
7
s^2 − 16
[
(a)6cosh4t (b)
7
4
sinh4t
]
- (a)
15
3 s^2 − 27
(b)
4
(s− 1 )^3
[
(a)
5
3
sinh3t (b)2ett^2
]
- (a)
1
(s+ 2 )^4
(b)
3
(s− 3 )^5
[
(a)
1
6
e−^2 tt^3 (b)
1
8
e^3 tt^4
]