Higher Engineering Mathematics

Laplace transforms

66

Inverse Laplace transforms

66.1 Definition of the inverse Laplace

transform

If the Laplace transform of a functionf(t)isF(s),

i.e.L{f(t)}=F(s), thenf(t) is called theinverse

Laplace transform of F(s) and is written as

f(t)=L−^1 {F(s)}.

For example, sinceL{ 1 }=

1

s

thenL−^1

{

1

s

}

= 1.

Similarly, sinceL{sinat}=

a

s^2 +a^2

then

L−^1

{

a

s^2 +a^2

}

=sinat, and so on.

66.2 Inverse Laplace transforms of

simple functions

Tables of Laplace transforms, such as the tables in

Chapters 64 and 65 (see pages 628 and 632) may be

used to find inverse Laplace transforms.

However, for convenience, a summary of inverse

Laplace transforms is shown in Table 66.1.

Problem 1. Find the following inverse Laplace

transforms:

(a)L−^1

{

1

s^2 + 9

}

(b)L−^1

{

5

3 s− 1

}

(a) From (iv) of Table 66.1,

L−^1

{

a

s^2 +a^2

}

=sinat,

HenceL−^1

{

1

s^2 + 9

}

=L−^1

{

1

s^2 + 32

}

=

1

3

L−^1

{

3

s^2 + 32

}

=

1

3

sin 3t

Table 66.1 Inverse Laplace transforms

F(s)=L{f(t)} L−^1 {F(s)}=f(t)

(i)

1

s

1

(ii)

k

s

k

(iii)

1

s−a

eat

(iv) a

s^2 +a^2

sinat

(v)

s

s^2 +a^2

cosat

(vi)

1

s^2

t

(vii)

2!

s^3

t^2

(viii)

n!

sn+^1

tn

(ix)

a

s^2 −a^2

sinhat

(x)

s

s^2 −a^2

coshat

(xi)

n!

(s−a)n+^1

eattn

(xii)

ω

(s−a)^2 +ω^2

eatsinωt

(xiii)

s−a

(s−a)^2 +ω^2

eatcosωt

(xiv)

ω

(s−a)^2 −ω^2

eatsinhωt

(xv)

s−a

(s−a)^2 −ω^2

eatcoshωt

(b) L−^1

{

5

3 s− 1

}

=L−^1

⎧

⎪⎪

⎨

⎪⎪

⎩

5

3

(

s−

1

3

)

⎫

⎪⎪

⎬

⎪⎪

⎭

=

5

3

L−^1

⎧

⎪⎪

⎨

⎪⎪

⎩

1

(

s−

1

3

)

⎫

⎪⎪

⎬

⎪⎪

⎭

=

5

3

e

1

3 t

from (iii) of Table 66.1