Higher Engineering Mathematics

INTRODUCTION TO LAPLACE TRANSFORMS 629

K

64.5 Worked problems on standard

Laplace transforms

Problem 1. Using a standard list of

Laplace transforms determine the following:

(a)L

{

1 + 2 t−

1

3

t^4

}

(b)L{5e^2 t−3e−t}.

(a)L

{

1 + 2 t−

1

3

t^4

}

=L{ 1 }+ 2 L{t}−

1

3

L{t^4 },

from equations (2) and (3)

=

1

s

+ 2

(

1

s^2

)

−

1

3

(

4!

s^4 +^1

)

,

from (i), (vi) and (viii) of Table 64.1

=

1

s

+

2

s^2

−

1

3

(

4. 3. 2. 1

s^5

)

=

1

s

+

2

s^2

−

8

s^5

(b) L{5e^2 t−3e−t}= 5 L(e^2 t)− 3 L{e−t},

from equations (2) and (3)

= 5

(

1

s− 2

)

− 3

(

1

s−(−1)

)

,

from (iii) of Table 64.1

=

5

s− 2

−

3

s+ 1

=

5(s+1)−3(s−2)

(s−2)(s+1)

=

2 s+ 11

s^2 −s− 2

Problem 2. Find the Laplace transforms of:

(a) 6 sin 3t−4 cos 5t (b) 2 cosh 2θ−sinh 3θ.

(a)L{6 sin 3t−4 cos 5t}

= 6 L{sin 3t}− 4 L{cos 5t}

= 6

(

3

s^2 + 32

)

− 4

(

s

s^2 + 52

)

,

from (iv) and (v) of Table 64.1

=

18

s^2 + 9

−

4 s

s^2 + 25

(b)L{2 cosh 2θ−sinh 3θ}

= 2 L{cosh 2θ}−L{sinh 3θ}

= 2

(

s

s^2 − 22

)

−

(

3

s^2 − 32

)

from (ix) and (x) of Table 64.1

=

2 s

s^2 − 4

−

3

s^2 − 9

Problem 3. Prove that

(a)L{sinat}=

a

s^2 +a^2

(b)L{t^2 }=

2

s^3

(c)L{coshat}=

s

s^2 −a^2

.

(a) From equation (1),

L{sinat}=

∫∞

0

e−stsinatdt

=

[

e−st

s^2 +a^2

(−ssinat−acosat)

]∞

0

by integration by parts,

=

1

s^2 +a^2

[e−s(∞)(−ssina(∞)

−acosa(∞))−e^0 (−ssin 0

−acos 0)]

=

1

s^2 +a^2

[(0)−1(0−a)]

=

a

s^2 +a^2

(provideds>0)

(b) From equation (1),

L{t^2 }=

∫∞

0

e−stt^2 dt

=

[

t^2 e−st

−s

−

2 te−st

s^2

−

2e−st

s^3

]∞

0

by integration by parts twice,