Higher Engineering Mathematics, Sixth Edition

588 Higher Engineering Mathematics

Problem 1. Determine (a)L{ 2 t^4 e^3 t}

(b)L{4e^3 tcos5t}.

(a) From (i) of Table 62.1,

L{ 2 t^4 e^3 t}= 2 L{t^4 e^3 t}= 2

(

4!

(s− 3 )^4 +^1

)

=

2 ( 4 )( 3 )( 2 )

(s− 3 )^5

=

48

(s−3)^5

(b) From (iii) of Table 62.1,

L{4e^3 tcos5t}= 4 L{e^3 tcos5t}

= 4

(

s− 3

(s− 3 )^2 + 52

)

=

4 (s− 3 )

s^2 − 6 s+ 9 + 25

=

4(s−3)

s^2 − 6 s+ 34

Problem 2. Determine (a)L{e−^2 tsin3t}

(b)L{3eθcosh4θ}.

(a) From (ii) of Table 62.1,

L{e−^2 tsin3t}=

3

(s−(− 2 ))^2 + 32

=

3

(s+ 2 )^2 + 9

=

3

s^2 + 4 s+ 4 + 9

=

3

s^2 + 4 s+ 13

(b) From (v) of Table 62.1,

L{3eθcosh 4θ}= 3 L{eθcosh 4θ}=

3 (s− 1 )

(s− 1 )^2 − 42

=

3 (s− 1 )

s^2 − 2 s+ 1 − 16

=

3 (s− 1 )

s^2 − 2 s− 15

Problem 3. Determine the Laplace transforms of

(a) 5e−^3 tsinh2t(b) 2e^3 t(4cos2t−5sin2t).

(a) From (iv) of Table 62.1,

L{5e−^3 tsinh2t}= 5 L{e−^3 tsinh2t}

= 5

(

2

(s−(− 3 ))^2 − 22

)

=

10

(s+ 3 )^2 − 22

=

10

s^2 + 6 s+ 9 − 4

=

10

s^2 + 6 s+ 5

(b) L{2e^3 t(4cos2t−5sin2t)}

= 8 L{e^3 tcos2t}− 10 L{e^3 tsin2t}

=

8 (s− 3 )

(s− 3 )^2 + 22

−

10 ( 2 )

(s− 3 )^2 + 22

from (iii) and (ii) of Table 62.1

=

8 (s− 3 )− 10 ( 2 )

(s− 3 )^2 + 22

=

8 s− 44

s^2 − 6 s+ 13

Problem 4. Show that

L

{

3e−

1

2 xsin^2 x

}

=

48

( 2 s+ 1 )( 4 s^2 + 4 s+ 17 )

Since cos2x= 1 −2sin^2 x,sin^2 x=

1

2

( 1 −cos2x).

Hence,

L

{

3e−

1

2 xsin^2 x

}

=L

{

3e−

1

2 x^1

2

( 1 −cos 2x)

}

=

3

2

L

{

e−

1

2 x

}

−

3

2

L

{

e−

1

2 xcos2x

}

=

3

2

⎛

⎜

⎜

⎝

1

s−

(

−

1

2

)

⎞

⎟

⎟

⎠−

3

2

⎛

⎜

⎜

⎜

⎝

(

s−

(

−

1

2

))

(

s−

(

−

1

2

)) 2

+ 22

⎞

⎟

⎟

⎟

⎠

from (iii) of Table 61.1 (page 584) and (iii)

of Table 62.1 above,

=

3

2

(

s+

1

2

)−

3

(

s+

1

2

)

2

[(

s+

1

2

) 2

+ 22

]

=

3

2 s+ 1

−

6 s+ 3

4

(

s^2 +s+

1

4

+ 4

)