SRI VENKATESWARA COLLEGE OF ENGINEERING
DEPARTMENT OF APPLIED MATHEMATICS
WORK SHEET- MA6251- MATHEMATICS – II
UNIT III–LAPLACE TRANFORM
Part A
1. State the sufficient conditions for the existence of Laplace transform.
2
2. Give two examples for which the Laplace transform do not exist.
[Ans: et , tan t ]
3. Give an example of a function such that it has Laplace transform but it is not satisfying the
continuity condition.
[Ans: 1 t ]
4. State & prove first shifting theorem.
5. If L( f (t ))  F ( s) , prove that L( f (at ))  (1 a) F  s a  .
6. If L( f (t ))  F ( s) , prove that L( f (t a))  aF  as  .
7. Define Heaviside’s unit step function; find its Laplace transform.
8. Define unit impulse (Dirac-delta) function.
9. State Initial Value & Final Value Theorems.
10. State convolution theorem.
11. Define inverse Laplace transform as contour integral.
12. Find the Laplace transform of the following functions:
(i)t sin 2t (ii) cos 2 3t (iii)sin t cos3t (iv)sin 3 4t
2
1
11
s 
6
4s
 2
[Ans:(i) 2
(ii)   2
(iv) 2
]
 (iii) 2
2
s  16 s  4
2  s s  36 
( s  1)( s 2  9)
( s  4)
13. Find the inverse Laplace transform of the following functions:
5t 2
et t 3
2s 2  4s  5
1
s
[Ans(i) 2  4t 
(ii)
(iii) e2t (1  2t ) ]
(i)
(ii)
(iii)
3
4
2
2
6
s
( s  1)
( s  2)

14. If L  f (t )  
s2
, find  f (t )dt .
s2  4
0
15. If L( f (t )) 
1
, find f (0) & f ()
s( s  a)
[Ans:1/2]
[Ans: 0, 1/a]
Part-B
I. State & prove the following results:
(1) Second Shifting Theorem (2) Initial Value Theorem (3) Final Value Theorem
II.Derive the Laplace transform of a periodic function f (t) with the period P.
III.Find the Laplace transform of the unit impulse (Dirac-Delta) function.
IV.Find the Laplace transform of the following functions:
s3
6( s  4)
(1) cosh at cos at (2) te 4t sin 3t
[Ans: (1) 4
(2) 2
]
4
s  4a
( s  8s  25) 2
e  at  e  bt
2s( s 2  3)
 sb 
(3) t 2 e t cos t
(4)
[Ans: (3) 2
(4) log 
]
3
t
( s  1)
sa
(5)
cos 2t  cos 3t
t
(7) e
4 t
t
 t sin 3tdt
0
e3t sin 2t
t
t
sin 3t
dt
(8) e 2t 
t
0
(6)
s 2  b2
 s2
(6) cot 1 
]
2
2
s a
 3 
1
6
s2
cot 1 
[Ans:(7) 2
(8)
]
2
s2
( s  8s  25)
 3 
[Ans: (5) log
 cos t 
 41s
V.Given L(sin t ) 
e .
e , show that L 

s
s 2s
t 

VI.Find the Laplace transform of the following functions:
t , 0  t  a
(1) The rectangular-wave function f (t )  
and f (t  2a)  f (t )
 2a  t , a  t  2a
a sin t , 0  t  

 and f (t  2 )  f (t )
(2)The half-sine wave rectifier function f (t )  

0,   t  2




k , 0  t  a
(3) The square-wave function f (t )  
and f (t  2a)  f (t )
  k , a  t  2a
(4)The saw-tooth wave function f (t )  kt p, for 0  t  p and f (t  p)  f (t )
 1
[Ans: (1)
1
4s
a
1
k
k
ke  s
 as 
 as 
tanh
tanh
(2)
(3)
(4)
]

 
 
s2
s 2 s 1  e  s 
 2
 2
 s 2  2 1  e  s /  s
VII.Find the inverse Laplace transform of the following functions:
s2
s 1
s2
se s
 s 1
1 k
(1)
(2) cot
(3) 2
(4) log 
(6) 4
 (5)
s
s  4a 4
( s  1)( s 2  4s  13)
( s  4s  5) 2
( s  3)5
 s 1 
 
1 3(t 1)
 sin kt
2sinh t
e 2t t sin t
3
4
e
4(
t

1)

3(
t

1)
u
(
t
)
(2)
(3)
(4)


1
24
t
t
2
1 2t
1
(5) e (cos 3t  2sin 3t )  e t  (6)  cos at sinh at  sin at cosh at  ]
5
2a
VIII.Verify Initial & Final Value Theorems for the following functions:
1. f (t )  1  et (sin t  cos t )
[Ans:(1)
2. f (t )  et (t  2)2
IX.Find the inverse Laplace transform of the following functions using convolution theorem:
s
2
s2
(1) 2
(2)
(3) 2
2 2
2
(s  a )
( s  1)( s  4)
( s  4s  13) 2
4
s2
(4) 2
(5) 2
( s  2s  5) 2
( s  a 2 )(s 2  b2 )
t sin at 2   t
a sin at  b sin bt
sin 2t  e 2t t sin 3t 1  t
2)  e  cos 2t 
4) e (sin 2t  2t cos 2t ) 5)
]
 3)
2a
4
a 2  b2
5
2 
6
X. Solve the following differential equations using Laplace transform:
4
1
1. y '' 9 y  cos 2t , y (0)  1, y ( / 2)  1
[Ans: y (t )   cos 3t  sin 3t   cos 2t ]
5
5
t 2 et
t
t
t
2. y '' 2 y ' y  e , y(0)  2, y '(0)  1
[Ans: y (t )  te  2e 
]
2
3. y '' y ' 2 y  3cos 3t  11sin 3t , y(0)  0, y '(0)  6
[Ans: y  sin 3t  e2t  et ]
4. x '' 3x ' 2 x  2(t 2  t  1), x(0)  2, x '(0)  0
[Ans: x  t 2  2t  3  e 2t ]
[Ans:1)
5. x "  2 x '  x  t 2 e t , x(0)  2, x1 (0)  3
6. y "  3 y '  4 y  2e t , y(0)  1  y1 (0)
7. y"2 y'5 y  e t sin t , y(0)  0, y' (0)  1
[Ans: x  e t [(t 4  12t  24) / 12] ]
[Ans: y  (12e 4t  13e t  10te t ) / 25 ]
[Ans: y  e t (sin t  sin 2t ) / 3 ]
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