Higher Engineering Mathematics, Sixth Edition

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:

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)2cos2t

]

3. (a)

1

s^2 + 25

(b)

4

s^2 + 9

[

(a)

1

5

sin5t (b)

4

3

sin3t

]

4. (a)

5 s

2 s^2 + 18

(b)

6

s^2

[

(a)

5

2

cos3t (b) 6 t

]

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)6cosh4t (b)

7

4

sinh4t

]

7. (a)

15

3 s^2 − 27

(b)

4

(s− 1 )^3

[

(a)

5

3

sinh3t (b)2ett^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

]