Higher Engineering Mathematics, Sixth Edition

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:

1. (a) 2te^2 t(b)t^2 et
   [
   (a)

2

(s− 2 )^2

(b)

2

(s− 1 )^3

]

2. (a) 4t^3 e−^2 t(b)

1

2

t^4 e−^3 t

[

(a)

24

(s+ 2 )^4

(b)

12

(s+ 3 )^5

]

3. (a) etcost(b) 3e^2 tsin2t
   [
   (a)

s− 1

s^2 − 2 s+ 2

(b)

6

s^2 − 4 s+ 8

]

4. (a) 5e−^2 tcos3t(b) 4e−^5 tsint
   [
   (a)

5 (s+ 2 )

s^2 + 4 s+ 13

(b)

4

s^2 + 10 s+ 26

]

5. (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

)

⎤

⎥

⎥

⎦

6. (a) etsinht(b) 3e^2 tcosh4t
   [
   (a)
   1
   s(s− 2 )

(b)

3 (s− 2 )

s^2 − 4 s− 12

]

7. (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 )

]

8. (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.