Higher Engineering Mathematics

640 LAPLACE TRANSFORMS

(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−tsinh 2t,

from (xiv) of Table 66.1

(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 tcosh 3t+L−^1

⎧

⎪⎨

⎪⎩

5

3

(3)

(s−2)^2 − 32

⎫

⎪⎬

⎪⎭

from (xv) of Table 66.1

=4e^2 tcosh 3t+

5

3

e^2 tsinh 3t,

from (xiv) of Table 66.1

Now try the following exercise.

Exercise 235 Further problems on inverse

Laplace transforms of simple functions

Determine the inverse Laplace transforms of the

following:

1. (a)

7

s

(b)

2

s− 5

[(a) 7 (b) 2e^5 t]

2. (a)

3

2 s+ 1

(b)

2 s

s^2 + 4

[

(a)

3

2

e−

1

2 t (b) 2 cos 2t

]

3. (a)

1

s^2 + 25

(b)

4

s^2 + 9

[

(a)

1

5

sin 5t (b)

4

3

sin 3t

]

4. (a)

5 s

2 s^2 + 18

(b)

6

s^2

[

(a)

5

2

cos 3t (b) 6t

]

5. (a)

5

s^3

(b)

8

s^4

[

(a)

5

2

t^2 (b)

4

3

t^3

]

6. (a)

3 s

1

2

s^2 − 8

(b)

7

s^2 − 16

[

(a) 6 cosh 4t (b)

7

4

sinh 4t

]

7. (a)

15

3 s^2 − 27

(b)

4

(s−1)^3

[

(a)

5

3

sinh 3t (b) 2 ett^2

]

8. (a)

1

(s+2)^4

(b)

3

(s−3)^5

[

(a)

1

6

e−^2 tt^3 (b)

1

8

e^3 tt^4

]

9. (a)

s+ 1

s^2 + 2 s+ 10

(b)

3

s^2 + 6 s+ 13

[

(a) e−tcos 3t (b)

3

2

e−^3 tsin 2t

]

10. (a)

2(s−3)

s^2 − 6 s+ 13

(b)

7

s^2 − 8 s+ 12

[

(a)2e^3 tcos 2t (b)

7

2

e^4 tsinh 2t

]

11. (a)

2 s+ 5

s^2 + 4 s− 5

(b)

3 s+ 2

s^2 − 8 s+ 25

⎡

⎢

⎣

(a) 2e−^2 tcosh 3t+

1

3

e−^2 tsinh 3t

(b) 3e^4 tcos 3t+

14

3

e^4 tsin 3t

⎤

⎥

⎦

66.3 Inverse Laplace transforms using

partial fractions

Sometimes the function whose inverse is required

is not recognisable as a standard type, such as those

listed in Table 66.1. In such cases it may be possible,

by using partial fractions, to resolve the function into