MATHEMATICS-II
UNIT-V
LAPLACE TRANSFORMS
PART B QUESTIONS & ANSWERS
1 .Verify Initial Value theorem for f (t)  et sint .
Solution:
Initial Value theorem: lim f (t)  lim sF(s)
t 0
s
Now,
lim f (t)  lim e  t sin t   e0 sin 0  1* 0  0

t 0 
t 0
 1 
lim sF(s)  lim s.L e  t sin t   lim s 

 s  s 2  1 
s
s 
s s 1




1
1
 lim s 
  lim s 

2
s   (s  1)  1 
s  s 2  2s  2 
s
1
1
 lim
 lim
 0
2 2  s 
2 2 
s  2 
s 1   
s 1   
s s2 
s s2 


 LHS  RHS.
Hence verified.
2. Find the inverse laplace transform of
 s  1
log 
.
 s 
Solution:
  s  1 
1  1 d
1 1  d
 s  1 

L1  log 
    t  L ds log  s     t L  ds  log(s  1)  logs  
s





 

1
1
1
1  e t
 1
  L1 
    e t  1 

t
t
t
s  1 s 
  s  1   1  e t
L1  log 
 
t
  s 
3.Find L[3e5t  5cost] .
Solution:
L[3e5t  5cost]  3L[e5t ]  5L[cost]
3
1
s
5
2
s5
s 1
SNSCT-DEPT OF MATHEMATICS
1
MATHEMATICS-II
1 s
L[F(at)]  F  
a a
4. If L[F(t)]=F(s),prove that
Proof:
By definition, we have

L[f (at)]   e  stf (at)dt
0
x
dx
put at  x  t  ;dt 
a
a

L[f (at)]   e  s(x / a)f (x)
0
dx
a
 s
1   a  x

e
f (x)dx
a
0
 s
1   a  t

e
f (t)dt
a
0
1 s
L[f (at)]  F  
a a
5. Find
L[eat sinbt]
L[eat sinbt]  L[sinbt]ss  a
 b 


 s2  b 2  ss  a


6. Find
b
(s  a)2  b 2
b
2
2
s  a  2sa  b 2
 1 
L1 

 (s  2)3 
 1 
1
L1 
  e2t L1  
3
 s3 
 (s  2) 
 t2 
 e 2t  
 2! 
 1 
t2
L1 
  e2t
3
2!
 (s  2) 
7.. If
L[f (t)]  F(s)
and
f (t  a),t  a
g(t)  
0,t  a

then prove that
a

0
a
L[g(t)]   est g(t)dt   est g(t)dt   est g(t)dt
0
a

0
a
  est 0dt   estf (t  a)dt
SNSCT-DEPT OF MATHEMATICS
2
L[g(t)]  easF(s)
MATHEMATICS-II
Put
t  a  x;t  a,x  0
dt  dx;t  ,x  


L[g(t)]   es(x a)f (x)dx   esxesaf (x)dx
0
0

 e sa  e sxf (x)dx
0

 e sa  e st f (t)dt
0
 e saF(s)
8. Find
L[te2t cos2t]
L[te2t cos 2t]  


d
L[te2t cos 2t]
ds
d
  L[cos 2t]ss  a
ds
d  s 
 

ds  s 2  22  ss  a
 (s 2  4)(1)  s(2s) 

 


(s 2  4)2
ss  a
 s 2  4  2s 2 

 
 (s 2  4)2 
ss  a
 4  s2 

 
 (s 2  4)2 
ss  a
  (s 2  4) 
 (s 2  4) 


 

 (s 2  4)2 
 (s 2  4)2 
ss  a
ss  a
 (s  2)2  4) 
s 2  2s


2
2
2
 [(s  2)  4]  [s  2s  s]2
Hence
L[te2t cos 2t] 
s2  2s
[s2  2s  s]2

9.
Prove that  te3t sintdt  3
50
0

 te
0
3t


sintdt    est (t sint)dt 


0
s 3
d
L[sin t]s  3
ds
 (s 2  1)(0)  1(2s) 
d  1 

 
 

ds  s 2  1  s  3


(s 2  1)2
s 3
 L[t sin t]s  3  
 2s 
 2s 
6
3
 





2
2
2
2
2
50
 (s  1)  s  3  (s  1)  s  3 (10)
SNSCT-DEPT OF MATHEMATICS
3
MATHEMATICS-II

Hence  te3t sintdt  3
50
0
10. Using Laplace Transforms of derivatives find
By Laplace Transforms of derivatives,
L[eat ]
L{f (t)}  sL{f (t)}  f (0)
L{f (t)}  s 2L{f (t)}  sf (0)  f (0)
L[eat ] ;
f (t)e at  ,f (0)e0  1
f (t)e at ( a)
 L[f (t)]  sL[f (t)]  f (0)
 L[aeat ]  sL[eat ]  1
s
a
 1 
 s
1 s a 1 s a
sa
a
L[ae at ] 
sa
a
 aL[e at ] 
sa
1
 at
L[e ] 
sa
11. Verify the initial value theorem for 3+4 cos2t
Theorem:
lim f (t)  lim sF(s)
t 0
s


lim (3  4cos 2t)  3  4  lim cos 2t   3  4[cos0]  7
 t 0

t 0
lim sF(s) = lim sL[3  4cos 2t]  lim s[L(3)  4L(cos 2t)]
s 
s
s
 1
s 
 lim s[3L(1)  4L(cos 2t)]  lim s  3.  4

2
s
s
s 
s  4




 1  

s2 


  lim  3  4 
 lim  3  4
1  4 
s  
s 2  4  s 


s 2  




 1 
 1 
  3  4
 3  4 lim 

s   1  4 
1  4 



s 2 
 1 
 3  4
7
1  0 
L.H.S = R.H.S
SNSCT-DEPT OF MATHEMATICS
4
MATHEMATICS-II
 1 
L

 t 
12. Find
 1

 1
1  2 
1   2  1
 1 


L
L t



 
   1 1 
 t 


 s 2 
 1 


1  2 
1  
1




1
  12 
  12 
s  s 2
 s 
1
 1 
L
 1
 t 
s 2
13. State Initial value theorem on Laplace Transforms.
If the Laplace transforms of f(t) and f ‘(t) exist and
then lim[f (t)]  lim [sF(s)]
t 0
14. Find .
L[f (t)]  F(s)
s
 4s  13 
L1 

 s2  5 
 4s  13 
1  4s 
1  13 
L1 
L  2
L  2

2
 s 5 
s  5
s  5




s
1
 4L1 
  13L1 

2
2
2
2
 s  ( 5) 
 s  ( 5) 

13 1 
5
 4cos 5t 
L 

5
 s 2  ( 5)2 
13
 4cos 5t 
sin 5t
5
15. State convolution theorem on Laplace Transforms.
If f(t) and g(t) are two functions defined for
t0,
then
L[(f * g)(t)]  L[f (t)].L[g(t)]
ie.,
16. Find
L[(f * g)(t)]  F(s).G(s)
where
 cosat  cosbt 
L

t



 cosat  cosbt 
L
   L  cosat  cosbt ds
t

 s
SNSCT-DEPT OF MATHEMATICS
5
L[f (t)]  F(s)
and
L[g(t)]  G(s)
MATHEMATICS-II



s
s

 
 
 L cosat  L cosbt ds    s2  a2    s2  b2  ds
s
s



1
1
1

  log(s 2  a 2 )  log(s 2  b 2 )   log(s 2  a 2 )  log(s 2  b 2 ) 


2
2
s
2
s


1   s2  a2  
1   s 2 (1  a 2 / s 2 )  
    log 

  log 
2   s 2  b 2  
2   s 2 (1  b 2 / s 2 )  

s

s

 s 2  a 2  
1   1  a2 / s2  
1 
    log1  log 

  log 
 s 2  b 2  
2   1  b 2 / s 2  
2



s

 s 2  a 2   1   s 2  b 2  
1 
 

   0  log 
log 
 s 2  b 2   2   s 2  a 2  
2




 
17. Find the inverse Laplace transform of
Now
s2
2
(s  3)(s  4)



s2


2
 (s  3)(s  4) 
A
Bs  c

s  3 s2  4
s  2  A(s2  4)  (Bs  c)(s  3)
Put s = -3
1  A(9  4)   B( 3)  c  (0)  A 
1
13
Put s = 0
2  4A  c(3)  c 
10
13
Put s = 1
3  A(5)   B  c  (4)  B 

s2
2
(s  3)(s  4)

1
13
1
1
s
10
1


2
2
13 (s  3) 13(s  4) 13 (s  4)


 1
s2
1
s
10
1 
L1 


  L1  

2
2
2
 (s  3)(s  4) 
 13 (s  3) 13(s  4) 13 (s  4) 

 10 1  1 
1 1  1  1 1  s
L 
 L 

 L  2
13
 (s  4)  13
 (s 2  4) 
 (s  3)  13
1  st 1
10 sin 2t
e
 cos 2t 
13
13
13 2
1
 cos 2t  5sin 2t  e  st 

13 

18. Find the inverse Laplace transform of
 s5 
log 

 s2  9 
  s  5 
1 1
1
L1  log 
   L [F(s)]   L [F(s)]
t
  s 2  9  
SNSCT-DEPT OF MATHEMATICS
6
MATHEMATICS-II
Now,
 s5 
2
F(s)  log 
  log(s  5)  log(s  9)
 s2  9 
1
1
F(s) 

.2s
2
s5 s 9
 1
 s 
2s 
1  1 
L1  F(s)  L1 

 2L1 
L 


2
s  5
s  5 s  9
 s2  9 
 e5t  2cos 3t
Now,
1
1
2cos 3t  e5t
L1  F(s)   L1  F(s)   e5t  2cos 3t  

t
t
t
19. If
L[f (t)] 

s2
, find the value of  f (t)dt
s2  4
0


 st

  e f (t)dt 
0



s0
 s2 
2 1
 L[f (t)s  0  
 

 s2  4 s  0 4 2
20. State the first Shifting theorem on Laplace transforms
If
L[f (t)]  F(s)
then
(i)L[eatf (t)]  F(s  a)
(ii)L[e atf (t)]  F(s  a)
21. State and prove second Shifting theorem on Laplace transforms.
If
L[f (t)]  F(s)
L[f(t  a)u(t  a)]  easF(s) .
then
Proof:

L[f (t  a)u(t  a)]   estf (t  a)u(t  a)dt
0
a

0
a
  est (0)dt   estf (t  a)dt
[ By defn. of unit step function]


e
 s(u  a)
f (u)du
put,t  a  u
0
e
 as

e
 su
f (u)du
0
tau0
tu
 e  as F(s)
SNSCT-DEPT OF MATHEMATICS
dt  du
7
MATHEMATICS-II
22. Define unit impulse function
The unit impulse function is defined by
,t  a
(t  a)  
0,t  a

such that 
(t  a)dt  1 .It
denoted by
(t  a) .
exists only at t = a at which it is infinitely great and is

23.State the Laplace transforms of periodic function with period transforms.
The Laplace transforms of a periodic function f(t) with period ‘p’ given by,
L[f (t)] 
24. Find
1
T
e
1  e Ts 0
st
f (t)dt
L sinh 2 2t 


We know that
sinh2 2t 
1
cosh4t  1
2
cosh 2(2t)  1  2sinh 2 2t 


Now,
1
L sinh2 2t   L  cosh4t  1

 2
1
 L  cosh 4t   L 1
2
1 s
1
8
 
 
2  s 2  16 s  s(s 2  16)
25. Find
 1  es 

L1 
 s 
[ By Second Shifting property
We can write
L[easF(s)]  f(t  a)u(t  a) ]
 1  es 
 es 
1
1

  L1    L1 
  1  L1 e  s 
L1 
s
s

 s 
 s 
1

L1 es 
s

1
F(s) 
s
Consider
Here
1
L1  F(s)  L1    1  f (t)
s
1


 L1 e s   1.u(t  1)
s

 1  e s 
  1  u(t  1)
 L1 
 s 
SNSCT-DEPT OF MATHEMATICS
8
MATHEMATICS-II
 t
26. Evaluate  e
0
 t

e
0
sint
dt
t
using laplace transforms.



sint


dt    estf (t)dt 
 L[f (t)]s 1
t


0
s1

Now,
 sint 
L[f (t)]  L 
   L[sint]ds
 t  s


1
 s21 ds  tan
1
s

(s) 
s
 tan 1 (  )  tan 1 (s)

  tan 1 (s)  cot 1 s
2
L[f (t)]s 1 
 t

0
e
cot 1 ss1  4
sint

dt 
t
4
27. Find the Laplace transforms of impulse function.

L[(t  a)]  lim  (t  a)  lim  est  (t  a)dt
0
0
0

 lim
 0
 st
 e  (t  a)dt  lim
0
a 
 lim
 0
 0

a 

1  e st 


 0    s 
a 
 0
a 
 lim
 lim 
a
 0
a
e st   (t  a)dt  lim
a
e  st   (t  a)dt  lim

a
1   s(a  )  as 
e
e

 0 s 
1
 lim e  as  e  as  e  s 

 0 s 

 1  e a 
e as

lim 
s  0 
s


e as
se s e  as
lim

s  e  as
s  0 1
s
SNSCT-DEPT OF MATHEMATICS
1
e  st dt

9


a 
e  st   (t  a)dt
MATHEMATICS-II
28. Find
 e 2s 

L1 
 s  3 
1 

1   as

L1 e2s
  L e F(s)   f (t  a)u(t  a)
s

3


1
 1 
3t
F(s) 
 L1  F(s)  L1 
e
s3
s  3
1 

L1 e2s
 e3(t  2)u(t  2)
s  3 

29. Find
 sinat 
L
.
 t 

Hence show that


s
sint

dt 
t
2


a
 sinat 
L
  L[sinat]ds  
ds

2
2
 t  s
s s a

=
 1
 s 
s 
s
 a. tan 1     tan 1 ( )  tan 1     tan 1  
a
a
a
2
 s
 
a

s
a
 cot 1   (or)tan 1  
a
s
Put s = 0 and a = 1 we get


s
30.Find
sint

dt 
t
2
 1 
L1 

 s(s  a) 
t
t
 1 
1 
1
1 
L1 
   L [F(s)]dt   L 
dt
 s(s  a)  0
 s(s  a) 
0
t
t
 eat 
1
  [eat  1]
  eat dt  
a
a



0
0
31. Find


s2

L1 
2
2
2
 (s  a ) 




s2
s
  L1 s.
L1 

2
2
2
2
2
2
 (s  a ) 
 (s  a ) 

 d  t
d 1 
s

L 
   sinhat 
2
2
2
dt

 (s  a )  dt  2a

1
[at coshat  sinhat]
2a
SNSCT-DEPT OF MATHEMATICS
10
MATHEMATICS-II
UNIT-V
LAPLACE TRANSFORMS
PART - C QUESTIONS
1. Using convolution theorem find


1
L1 

 (s  1) (s  2) 
.

s2
L1 
 (s 2  a 2 ) (s 2  b 2 )
2. Using convolution theorem find

.

3. Find the inverse Laplace transform of the following function using convolution
Theorem 3 1 .
s  s  5


2
L1 
.
2
 (s  1)(s  4) 
4. Using convolution theorem find
5. Using Laplace transform method solve
y''  2y'  y  et given y(0)  2 and y' (0)  1 .
y''  4y'  4y  et
6. Solve the differential equation using Laplace transform
y(0)  0 and y' (0)  0 .
7. Solve by using Laplace transform
8. Solve
d 2y
dt
2
9
dy
 cos4t
dt
,
y''  3y'  2y  4
given that
given that
y(0)  2 , y' (0)  3 .

y(0)  1 , y    1 .
 2
9. Solve by Laplace transform the differential equation
d 2y
dt
2
9
dy
 18t
dt
given that

y(0)  0  y( ) .
2
10. Using Laplace transform method solve
11. Using Laplace transform method solve
y''  25y  10cos5t given y(0)  2 and y' (0)  0 .
d 2y
dt
12. Solve using Laplace transform,
2
dy
 2y  5sin t given
dt
that
y(0)  y' (0)  0 .
y''  3y'  2y  et given y(0)  1 and y' (0)  0 .
13. Using Laplace transform method solve
&
2
y''  2y'  3y  sint given
y'(0)  0
SNSCT-DEPT OF MATHEMATICS
11
that
y(0)  0
MATHEMATICS-II
14. Using Laplace transform method solve
&
y''  6y'  5y  e2t given
that
y(0)  0
y'(0)  1 .
15. Find the Laplace transforms of f(t) if
f (t)  et , 0  t  2
&
f (t)  f (t  2 )
16. Find the Laplace transform of the triangular wave function
.
 t 0t1
f (t)  
2  t 1  t  2
and
f (t)  f (t  2) .
17. Find the Laplace transforms of the periodic function
18. Find the Laplace transform of
period
 1 for 0  t  a
f (t)  
 1 for a  t  2a
sint when 0  t  
f (t)  
 0 when   t  2
and
f (t)
.
is periodic with
2 .
19. Find the Laplace transforms of rectangular wave function given by
T

 A for 0  t  2
f (t)  
  A for T  t  T

2
and
f (t  T)  f (t) .
20. Find the Laplace transform of the periodic function
 t 0t
.
f (t)  
 2  t   t  2
21. Find the Laplace transform of the triangular wave function
22. Find the Laplace transforms of
23. Find the Laplace transform of
period
2

 E for 0  t  
f (t)  
E for   t  2
and


sin t when 0  t  
f (t)  
 0 when   t  2



.
SNSCT-DEPT OF MATHEMATICS
12
 t 0tb
.
f (t)  
 2b  t b  t  2b
f (t  2 )  f (t) .
and
f (t)
is periodic with