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Laplace Transforms 1

Proof_01_(19-12-2014) Prof. A. Y. Shete … Mobile : 98693 38374

LAPLACE TRANSFORMS

Obtain Laplace transform of the following :

(i) 3sin t Ans:-

2 2

3 1 14 1 9s s

(ii) cos cos2 cos3t t t Ans:-

2 2 2

1

44 36 4 4 4 16

s s s

s s s s

(iii) 4cos t Ans:-

2 2

1 3 2

4 2 4 2 16

s s

s s s

(iv) cos (t + ) where and are constants.

Ans:-

2 2cos sins

s

(v) 1 sint Ans:-

2

1

2

1

4

s

s

(vi) Prove that

5

2 2 2

5!L sin

1 9 25t

s s s

(vii) Prove that

1/43/2

L sin2

s e s

(viii) Prove that

1/4cosL s t

e s t

Find the Laplace transform of f (t ) defined as

(i) t

f t k

, 0


2/17

2


Ans:-

2

2 1

4

s e

s

(iv) f (t ) =

1, 0 1

,1 40, 4

t

t

e t t

Ans:-

4 1 11

1 1

s s s e e e

s s s

Find Laplace transform of the following

(i) 22 t t e Ans:-

2

3

22 2 1

1s s

s

(ii) 3 sin3 sinh4t e t t Ans:-

2 2

3 1 1

2 2 10 14 58s s s s

(iii) sinh2s t t Ans:-

6 6

1 160

2 2s s

(iv)

2cos sin

t

t t

e Ans:-

2 2

1 2 1

2 2 4 2 1s s s

(v) 3 cosh 4t t e t Ans:-

2 21 1

2 7 2 1s s

(vi) 4 sin cosht e t t Ans:-

2 2

1 1

2 3 1 2 5 1s s

(vii)

cos2 sint t

et Ans:-

2 2

3 1

2 1 9 2 1 1s s

(viii) 4 sinh4t t Ans:-

3 3

1 1

4 4s s

(ix) sin 2t sin 4t sinh t

Ans:-

2 2 2 2

1 1 1 11

4 1 4 1 36 1 4 1 36

s s s s

s s s s

(x) sin 2t sin t cosh 2t

Ans:-

2 2 2 2

1 3 3 1 1

4 2 9 2 9 2 1 2 1s s s s

(xi) 31 t te Ans:-

2 3 41 3 6 6

1 2 3s s s s


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(xii) 4 cosh4t t Ans:-

5 5

1 112

4 4s s

(xiii) 3cosh4 sin3 t t t e Ans:-

2 2

1 3 3

21 9 7 9s s

Using second shifting property. Find

(i) L [g (t )], where

g (t ) = cos (t – ), t >

= 0, t 2

= 0, t < 2

Ans:- 23

2s e s

Change of scale property

If L f t f s , then

1

L s

f at f

a a

(i) If

2

22

8 125 25L

4

f at

s

, find L [ f (2t )]

Ans:-

2

22

4 16 12

16

s s

s

(ii) If 3

2L s f t e

s

, find L [ f (2t )]

Ans:- 23

8s

e s


4/17

4


Laplace transform of ( )n t f t

If L f t f s , then

L 1n

n n

n

d t f t f s

ds

Obtain Laplace transform of the following

(i) 2sint t Ans:-

2

2 22

41 1

24

s

s s

(ii) 2cost t Ans:-

2

2 22

41 1

24

s

s s

(iii) 2 sin 3t t Ans:-

3

32

18 3

9

s

s

(iv) sinh2

t at

a Ans:-

2

2 2

2as

s a

(v) 2 cost at Ans:-

2 2

32 2

2 3s s a

s a

(vi) 3 3t t e Ans:-

43

s

s

(vii) 3sint t Ans:-

2 2

2 2

3 3

2 1 2 9

s s

s s

(viii) sinat t e at Ans:-

2

2 2

2a s a

s a a

(ix) t sin 2t cosh t Ans:-

2 2

2 2

2 1 2 1

1 4 1 4

s s

s s

(x) t [3 sin 2t – 2 cos 2t ] Ans:-

2

22

2 12 8

4

s s

s

(xi) 1 sint t Ans:-

2

22

4 4 4 1

4 1

s s

s


5/17



(xii) L [ cos ( – )] Ans:-

2 2

22 2

cos 2 sins s

s

(xiii) 2

L sin2t t Ans:-

3 2 2

2

2 1

2 2 41

s s

s s s s

Laplace transform of ( )1

f t t

If L f t f s , then

1

L

s

f t f s ds t

Obtain Laplace transform of the following

(i) sin t

t Ans:- 1cot s

(ii)2sin 2t

t Ans:-

2

2

1 16log

4

s

s

(iii) cosh2 sint t

t

Ans:- 1

2

1 4tan

2 s s

(iv)

cos cosat bt

t

Ans:-

2 2

2 2

1

log2

s b

s a

(v)at bt e e

t

Ans:- log

s b

s a

(vi) sinh t

t Ans:-

1 1log

2 1

s

s

(vii) 1

, 0t e

t t

Ans:-

1log

s

s

(viii) 2 1sin3t e t t Ans:- 1 2cot3

s

(ix)2 sin2 cosht e t t

t

Ans:- 1 11 1 3

cot cot2 2 2

s s

(x)2sin t

t Ans:-

2

2

1 4log

4

s

s

Laplace transform of derivative

If L f t f s , then

L 0 f t s f s f


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6


where 0

0 limt

f f t

(i) Find sin

L d t

dt t

Ans:- 1

cot 1s s

(ii) Given, f (t ) = (t + 1), 0 ≤ t ≤ 2

= 3, t > 2, find L [ f (t )] and L f t

Ans:- 221 1

L 1 s f t e s s

and 21

L 1 s f t e s

(iii) Given, f (t ) = t , 0 t 3

= 6 , t > 3

Find L [ f (t )] and L f t

Ans:- 32 2

1 3 1L s f t e

s s s

and 31 1

L 3s f t e s s

(iv) Find Laplace transform of 1 cos2d t

dt t

Ans:-2 22

log s

s s

Laplace transform of an IntegralIf L f t f s , then

0

1L

k

f u du f s s


(i) 2 3

0

t u e u du Ans:-

4

6

2s s

(ii)

0

sin4t t t e dt t

Ans:-1cot s

s

(iii) 2

0

cost

u u du Ans:-

2

3 22

41 1

22 4

s

s s s

(iv)

0

sint

u t du

u Ans:-

1

22

1 cot

1

s

s s s

(v) 2

0

sin3t t e t t dt Ans:-

2

2

6

4 13s s


7/17



(vi) 3

0

sin4t

t t e t dt Ans:-

2

2

8 3

6 25

s

s s s

(vii)

0

sint

t t e dt

t

Ans:- 1cot 1s

s

(viii)

0

cosht

x x dx Ans:-

2

22

1

1

s

s s

(ix) 3

0 0 0

sint t t

t t dt Ans:-

22 2

2

1s s

Show that 1

L 1

e r f t s s


(i) 3t e erf t Ans:-

1

3 4s s

(ii) 2t e erf t Ans:-

1

2 3s s

(iii) 2t erf t Ans:-

3/22 2

2

1s s

(iv) 3t t e erf t Ans:-

2 3/2

3 7

2 3 2

s

s s

(v) If 1 1

L t s

, find L

t

Ans:-s

(vi) Find L c erf t Ans:- 1

1 1 1s s

Prove the following integrates using Laplace transform

(i) 3

0

2sin2

13

t e t dt

(ii) 3

0

3sin

50

t t e t dt

(iii) 3 2

0

62sin

1521

t t e t dt

(iv) 0

sin2 sin3 34t

t t dt t e


8/17

8


(v) 2 3

0

6sin

65

t e t dt

(vi)2

0

sinh sin

8

t e t t dt

t

(vii)2

0

sin 1log 5

4

t t e dt t

Evaluate the following integrates using Laplace transform

(i)

0

cos6 cos4t t dt

t

Ans:-

2log

3

(ii)2

0

sin3t

t t dt

e

Ans:- 12

169

(iii) 3 2

0

cos 2t e t t dt

Ans:-

281

5625

(iv) 3

0

sint t e t dt

Ans:- 0

(v) 2

0

sinht t e dt t

Ans:- 1

log 32

(vi)

0

t e erf t dt Ans:- 1

2

(vii) 3 00

J 4t t e t dt

where 0

2

1L J

1t

s

Ans:- 3

125

Inverse Laplace Transform

Evaluate the following

(i) 12

1L

9s

Ans:-

1sin 3

3t

(ii) 1 8

L 5 1s

Ans:- /58

5

t e

(iii)

22

1

5

7 1L

3

s

s

Ans:- 2 47 7 7

3 3 72t t

(iv) 12

3 7L

4 25

s

s

Ans:-

3 5 7 5cosh sin

4 2 10 2t h t


9/17



(v) 12

3 4 3L

7

s

s

Ans:-

4 33cos 7 sin 7

7t

Method of partial traction


(i)

21

2 2 2 2L

s

s a s b

Ans:-

2 2

1sin sina at b bt

a b

(ii) 21

3 2

1L

3 2

s

s s s

Ans:- 21

22 2

t t s e e

(iii)

1 2L

1 3

s

s s s

Ans:-

32

3 2 6

t t e e

(iv)

21

2 2

2 1L

1 4

s

s s

Ans:- 3sin2

sin2

t t

(v)

21

2 2

2 3L

2 2 2 5

s s

s s s s

Ans:- sin sin23

t e t t

(vi)

1

2L

1 4

s

s s

Ans:- cos2 2

sin25

t e t t

s s

(vii)

12

L 2 1

s

s s

Ans:- 2 22t t t e t e e

(viii)

1

2

5 3L

1 2 5

s

s s s

Ans:- 3cos2 sin2

2

t t t e e t e t

(ix) 14 2

L 1

s

s s

Ans:-

2 3sin sin

2 23

t t h

Formula : If ( ) ( )1L f t f s − = , then ( ) ( )1L at f s a e f t − − + =


(i)

1

2

2L

4 7

s

s s

Ans:- 2 24

cos 3 sin 33

t t e t e t

(ii)

15

3L

3s

Ans:-4 3

8

t t e


10/17

10


(iii)

15

L 5

s

s

Ans:-5 3

5 45

6 24

t t e t e t

(iv)

14

3 1L

1

s

s

Ans:-3

23

2 3

t t t e t e

(v) 12

2 3L

2 2

s

s s

Ans:- 2 cos sint t e t e t

(vi) 12

14 10L

49 28 13

s

s s

Ans:-

2

72 3 3

cos sin7 7 7

t e t t

Formula : If ( ) ( )1L f s f t − = , then ( ) ( ) ( )1L 1

d f s t f t

ds

− = −

Find the following

(i)2 2

12 2

L log s b

s a

Ans:-

2 cos cosat bt

t

(ii)

2

11

L log1

s

s s

Ans:- 1

1 2cost e t t

(iii) 1 1

L log1

s s k

s

where ‘ k ’ is a constant. Ans:-

2

2 2cosh sinht t

t t

(iv) 12

1L log 1

s

Ans:- 2 1 cos t

t

(v) 1 2 1

L log2 3

s

s

Ans:-

3 1

2 2t t

e e

t

(vi) 1 1

L log1

s

s

Ans:-

1sinh t

t

(vii) 1L log s m s n

Ans:-nt mt

e e t

(viii)2 2

12

L log s a

s

Ans:- 1 coshat

t

(ix) 1 1L cot s

Ans:- sint

t

(x) 1 1L tan5

a

Ans:- sinat

t

(xi) 1 1L tan 2s Ans:-2 sint e t

t


11/17



(xii) 1 12

2L tan

s

Ans:- sint t e e

t t

(xiii) 1 1 2

L tan3

s

Ans:-2 sin3t e t

t

(xiv) 1 1L 2 tanh s

Ans:- 2sinh t

t

Convolution Theorem

Using convolution theorem find

(i)

21

22

L

4

s

s

Ans:- 1

sin2 2 cos24

t t t

(ii)

1

22 2

L s

s a

Ans:- sin

2

t at

a

(iii)

12

1L

2 2s s

Ans:- 212

t t e

(iv)

122

1L

1s s

Ans:- 2 2t t te e t

(v)

21

22 2

L s

s a

Ans:- 1 1

cos sint at at t a

(vi)

21

2 2L

1 4

s

s s

Ans:- 1

2sin2 sin3

t t

(vii)

1

2

2

1L

4 13s s

Ans:-2 sin3

cos3

18 3

t e t t t

(viii)

21

22 2

L s

s a

Ans:- 1

cosh sinh2

at at at a

(ix)

21

22

2L

4 8

s

s s

Ans:-2 sin2

cos22 2

t e t t t

(x)

1

4 2L

13 36

s

s s

Ans:- 1

cos2 cos35

t t


12/17

12


Using Laplace transform technique, solve the following differential equations

(i)2

2

d y y t

dt , given y (0) = 1, 0 0y

Ans:- t + cost – sin t

(ii)2

2 4 8 1

d y dy y

dt dt , given y = 0 and 0

dy

dx at t = 0

Ans:-2

21 cos2 1 sin28 8 8

t t e t e t

(iii) 2D 2D 5 sint y e t where D d dt

and y (0) = 0, 0 1y

Ans:- sin sin23

t e t t

(iv) 23 3 t y t y t y t y t t e where y (0) = 0, 0 0y , 0 2y

Ans:-5 2

2120 2

t t t e t e

(v)2

2 2 3 t

d y dy y te

dt dt

, given y (0) = 4, 0 2y

Ans:-3

4 62

t t t e t e t e

(vi) 3 2D 2D 5D 0y , with y (0) = 0, 0 0y , 0 1y

Ans:- 1 sin2 cos2

5 10 5

t t t e t

(vii) 4 9x t x t t , where x (0) = 0 and 0 7x

Ans:- 9 19

sin24 8

t t

(viii) 3 2 2D 3D 3D 1 t x t e , given x (0) = 1, D[x (0)] = 0, 2D 0 2x where D d

dt

Ans:-5 2

60 2

t t t t e t t e e te

(ix)2

2 2 1

d y dy y

dt dt , given y (0) = 1 and 0 1y

Ans:- 1t te

(x)

0

2 sint

dy y y dt t

dt , where y (0) = 1

Ans:- 3 1 sin2 2

t t e e t t


13/17



(xi)2

2 9 18

d y y t

dt where y (0) = 0 and 0

2y

Ans:- 2t + sin 3t

(xii)2

2

2 2

d y dy t t

dt dt , given that y = 4 and 2

dy

dt at t = 0

Ans:-3

2 16

t t e

(xiii)2

2 9 cos 2

d y y t

dt , with y (0) = 1 and 1

2y

Ans:- 4 4 1

sin3 cos3 cos25 5 5

t t t

Laplace Transform of special functions

(I) Theory question (periodic function)

If f (t ) is periodic function with period T > 0 show that

T

T 0

1L

1

st

s f t e f t dt

e

, s > 0

Example

(i) Find Laplace transform of

T 0

2 T

T 2

k t

f t

k t

Where f (t ) = f (t + T)

Ans:- T

tanh4

k s

s

(ii) Find Laplace transform of

1 0

1 2

t b f t

b t b

Where f (t ) = f (t + 2b )

Ans:- 1

tanh2

bs

s

(iii) Find Laplace transform of

sin , 0

20,

a pt t p

f t

t p p

where 2

f t f t p


14/17

14


Ans:-

2 2 1s

p

ap

s p e

(iv) Find Laplace transform of

f (t ) = t , 0


15/17



(vii) Find the Laplace transform of sin f t p t , t 0

Ans:-

2 2P

2

s th

p s p

(II) Heaviside unit step function

Prove the following

(i) L Has e

t a s

(ii) 1

L H t s

(iii) If L f t f s then L H as f t a t a e f s

(iv) L H L as f t a t a e f t a

Find the following

(i) 3L sin H H2 2

t t t

Ans:-

3

2 22

1

1

s s s

e e s s

(ii) 2L H 2t t

Ans:- 23 2

2 4 4s e s s s

(iii) 2 3L 1 2 3 4 H 2te t t t

Ans:- 24 3 2

24 42 38 25s e s s s s

(iv) L [sint H (t – )]

Ans:-

21

1

s e s

(v)

4

L H 2t t

Ans:- 25 4 3 2

24 48 48 32 16s e s s s s s

(vi) 2 3L 1 2 H 1t t t t

Ans:-4 3 2

6 4 3 3s e s s s s

Find

(i)

15/2

L as e

s b


16/17

16


Ans:- 3/24

H3

b t a e t a t a

(ii)

31

3L

4

s e

s

Ans:-

2

4 3 2H

2

t t e t a

(iii)2

12

L 8 25

s e

s s

Ans:- 4 81

sin 3 6 H 23

t e t t

(iv)

4 31

5/2L

4

s e

s

Ans:- 4 3

4 32

43 H 3

3

t e e t t

(v)

1

2L

4

s e

s

Ans:- 1

sin2 H2

t t

(vi)

1

2

2L

3

s s e

s s

Ans:- 31 2 1

H 19 3 9

t t e t

(vii) Solve 2

2 4

d y y f t

dt , with condition f (t ) = H(t – 2), y (0) = 0, 0 1y

Ans:- 1 sin2

1 cos2 2 H 24 2

t y t t t

(III) Dirac-delta function

(Unit – impulse function)

Note : (i) L as t a e

(ii) L as f t t a e f a

(iii) 0

f t t a dt f a


(i) 2

0

sin 2t t e t t dt


17/17


Proof 01 (19 12 2014) Prof A Y Shete Mobile : 98693 38374

Ans:-2

4sin2

e

(ii) 0

log 3n m t t t dt

Ans:- 3 log3 n m

(iii)

0

cos23

t t dt

Ans:- 1

2

Find the following

(i) L sin2

4

t t

Ans:- 4s

e

(ii) 3 2L H 4 4t t t t

Ans:- 44 3 2

6 24 48 6416s e

s s s s

(iii) 2L sin2 44

t t t t

Ans:- 44 16

s s e e

(iv) 3L H 4 2t t t t

Ans:- 4 22

1 48s s e e

s s

(v) L sin2 2t t

Ans:- 2 sin4s e

(vi) 2L H 4 4t t t t

Ans:-4

2

2 1 4 10

s e s s

s