Higher Engineering Mathematics, Sixth Edition

584 Higher Engineering Mathematics

Table 61.1Elementary standard Laplace transforms

Function Laplace transforms

f(t) L{f(t)}=

∫∞

0 e

−stf(t)dt

(i) 1

1

s

(ii) k

k

s

(iii) eat

1

s−a

(iv) sinat

a

s^2 +a^2

(v) cosat

s

s^2 +a^2

(vi) t

1

s^2

(vii) t^2

2!

s^3

(viii) tn(n= 1 , 2 , 3 ,...)

n!

sn+^1

(ix) coshat

s

s^2 −a^2

(x) sinhat

a

s^2 −a^2

(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 61.1

=

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 61.1

=

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) 6sin3t−4cos5t (b)2cosh2θ−sinh3θ.

(a) L{6sin3t−4cos5t}

= 6 L{sin3t}− 4 L{cos5t}

= 6

(

3

s^2 + 32

)

− 4

(

s

s^2 + 52

)

,

from(iv)and(v)ofTable61.1

=

18

s^2 + 9

−

4 s

s^2 + 25

(b) L{2cosh2θ−sinh3θ}

= 2 L{cosh 2θ}−L{sinh3θ}

= 2

(

s

s^2 − 22

)

−

(

3

s^2 − 32

)

from(ix)and(x)ofTable61.1

=

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 (−ssin0

−acos 0)]