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
)