1. The document discusses Laplace transforms and provides definitions, properties, and examples. Laplace transforms take a function of time and transform it into a function of a complex variable s. 2. Key properties discussed include linearity, shifting theorems, and Laplace transforms of common functions like 1, t, e^at, sin(at), etc. Explicit formulas for the Laplace transforms of these functions are given. 3. Examples of calculating Laplace transforms of various functions are provided.

1. Laplace Transforms
Nirav B. Vyas
Department of Mathematics
Atmiya Institute of Technology and Science
Yogidham, Kalavad road
Rajkot - 360005 . Gujarat
N. B. Vyas Laplace Transforms

2. Laplace Transforms
De

3. nition:
Let f(t) be a function of t de

4. ned for all t 0 then Laplace
transform of f(t) is denoted by Lff(t)g or f(s) and is
de

5. ned as
L ff (t)g = f (s) =
Z1
0
estf (t) dt
provided the integral exists where s is a parameter ( real or
complex).
N. B. Vyas Laplace Transforms

6. Laplace Transforms
NOTATIONS:
The original functions are denoted by lowercase letters such
as f(t); g(t); :::
Laplace transforms by the same letters with bars such as
f(s)g(s); :::
N. B. Vyas Laplace Transforms

7. Linearity of the Laplace Transforms
Theorem 1:
If Lff (t)g = f (s) and Lfg (t)g = g (s) then for any constants a and b
Lfaf (t) + bg (t)g = aL ff (t)g + bL fg (t)g
N. B. Vyas Laplace Transforms

8. Linearity of the Laplace Transforms
Theorem 1:
If Lff (t)g = f (s) and Lfg (t)g = g (s) then for any constants a and b
Lfaf (t) + bg (t)g = aL ff (t)g + bL fg (t)g
Corollary 1:
Putting a = 0 and b = 0, we get L[0] = 0
N. B. Vyas Laplace Transforms

9. Linearity of the Laplace Transforms
Theorem 1:
If Lff (t)g = f (s) and Lfg (t)g = g (s) then for any constants a and b
Lfaf (t) + bg (t)g = aL ff (t)g + bL fg (t)g
Corollary 1:
Putting a = 0 and b = 0, we get L[0] = 0
Corollary 2:
Putting b = 0, we get L[af(t)] = aL[f(t)]
N. B. Vyas Laplace Transforms

10. Linearity of the Laplace Transforms
Theorem 1:
If Lff (t)g = f (s) and Lfg (t)g = g (s) then for any constants a and b
Lfaf (t) + bg (t)g = aL ff (t)g + bL fg (t)g
Corollary 1:
Putting a = 0 and b = 0, we get L[0] = 0
Corollary 2:
Putting b = 0, we get L[af(t)] = aL[f(t)]
Corollary 3:
L[a1f1 (t) + a2f2 (t) + ::: + anfn (t)]
= a1L[f1(t)] + a2L[f2(t)] + ::: + anL[fn(t)]
N. B. Vyas Laplace Transforms

11. Laplace Transforms of some elementary functions
1 L(1) =
1
s
N. B. Vyas Laplace Transforms

12. Laplace Transforms of some elementary functions
1 L(1) =
1
s
2 L(eat) =
1
s a
N. B. Vyas Laplace Transforms

13. Laplace Transforms of some elementary functions
1 L(1) =
1
s
2 L(eat) =
1
s a
cor.1 If a = 0 ) L(1) =
1
s
N. B. Vyas Laplace Transforms

14. Laplace Transforms of some elementary functions
1 L(1) =
1
s
2 L(eat) =
1
s a
cor.1 If a = 0 ) L(1) =
1
s
cor.2 L[eat] =
1
s + a
if s a
N. B. Vyas Laplace Transforms

15. Laplace Transforms of some elementary functions
1 L(1) =
1
s
2 L(eat) =
1
s a
cor.1 If a = 0 ) L(1) =
1
s
cor.2 L[eat] =
1
s + a
if s a
cor.3 L[cat] = L[eat log c] =
1
s a logc
if s a log c and c 0
N. B. Vyas Laplace Transforms

16. Laplace Transforms of some elementary functions
3 L[sinh at] =
a
s2 a2
N. B. Vyas Laplace Transforms

17. Laplace Transforms of some elementary functions
3 L[sinh at] =
a
s2 a2
4 L[cosh at] =
s
s2 a2
N. B. Vyas Laplace Transforms

18. Laplace Transforms of some elementary functions
3 L[sinh at] =
a
s2 a2
4 L[cosh at] =
s
s2 a2
5 L[sin at] =
a
s2 + a2
N. B. Vyas Laplace Transforms

19. Laplace Transforms of some elementary functions
3 L[sinh at] =
a
s2 a2
4 L[cosh at] =
s
s2 a2
5 L[sin at] =
a
s2 + a2
6 L[cos at] =
s
s2 + a2 ; s 0
N. B. Vyas Laplace Transforms

20. Laplace Transforms of some elementary functions
3 L[sinh at] =
a
s2 a2
4 L[cosh at] =
s
s2 a2
5 L[sin at] =
a
s2 + a2
6 L[cos at] =
s
s2 + a2 ; s 0
cor.1 L[sin t] =
1
s2 + 1
; s 0
N. B. Vyas Laplace Transforms

21. Laplace Transforms of some elementary functions
3 L[sinh at] =
a
s2 a2
4 L[cosh at] =
s
s2 a2
5 L[sin at] =
a
s2 + a2
6 L[cos at] =
s
s2 + a2 ; s 0
cor.1 L[sin t] =
1
s2 + 1
; s 0
cor.2 L[cos t] =
s
s2 + 1
; s 0
N. B. Vyas Laplace Transforms

22. Laplace Transforms of some elementary functions
7 L[tn] =
(n + 1)
sn+1
N. B. Vyas Laplace Transforms

23. Laplace Transforms of some elementary functions
7 L[tn] =
(n + 1)
sn+1
=
n!
sn+1 ; n = 0; 1; 2; :::
N. B. Vyas Laplace Transforms

24. Laplace Transforms of some elementary functions
7 L[tn] =
(n + 1)
sn+1
=
n!
sn+1 ; n = 0; 1; 2; :::
cor.1 If n = 0, L[1] =
1
s
N. B. Vyas Laplace Transforms

25. Laplace Transforms of some elementary functions
7 L[tn] =
(n + 1)
sn+1
=
n!
sn+1 ; n = 0; 1; 2; :::
cor.1 If n = 0, L[1] =
1
s
cor.2 If n =
1
2
N. B. Vyas Laplace Transforms

26. Laplace Transforms of some elementary functions
7 L[tn] =
(n + 1)
sn+1
=
n!
sn+1 ; n = 0; 1; 2; :::
cor.1 If n = 0, L[1] =
1
s
cor.2 If n =
1
2
L
t1
2
=
(1
)
2 1
s
2
=
r
s
N. B. Vyas Laplace Transforms

27. Laplace Transforms of some elementary functions
7 L[tn] =
(n + 1)
sn+1
=
n!
sn+1 ; n = 0; 1; 2; :::
cor.1 If n = 0, L[1] =
1
s
cor.2 If n =
1
2
L
t1
2
=
(1
)
2 1
s
2
=
r
s
N. B. Vyas Laplace Transforms

28. Examples of Laplace Transform
1 L
2t3 + e2t + t
4
3
N. B. Vyas Laplace Transforms

29. Examples of Laplace Transform
1 L
2t3 + e2t + t
4
3
2 L
A + B t
1
2 + C t
1
2
N. B. Vyas Laplace Transforms

30. Examples of Laplace Transform
1 L
2t3 + e2t + t
4
3
2 L
A + B t
1
2 + C t
1
2
3 L
eat 1
a
N. B. Vyas Laplace Transforms

31. Examples of Laplace Transform
1 L
2t3 + e2t + t
4
3
2 L
A + B t
1
2 + C t
1
2
3 L
eat 1
a
4 Lfsin(at + b)g
N. B. Vyas Laplace Transforms

32. Examples of Laplace Transform
1 L
2t3 + e2t + t
4
3
2 L
A + B t
1
2 + C t
1
2
3 L
eat 1
a
4 Lfsin(at + b)g
5 Lfsin 2t cos 3tg
N. B. Vyas Laplace Transforms

33. Examples of Laplace Transform
1 L
2t3 + e2t + t
4
3
2 L
A + B t
1
2 + C t
1
2
3 L
eat 1
a
4 Lfsin(at + b)g
5 Lfsin 2t cos 3tg
6 L
cos24t
N. B. Vyas Laplace Transforms

34. Examples of Laplace Transform
1 L
2t3 + e2t + t
4
3
2 L
A + B t
1
2 + C t
1
2
3 L
eat 1
a
4 Lfsin(at + b)g
5 Lfsin 2t cos 3tg
6 L
cos24t
7 L
cos32t
N. B. Vyas Laplace Transforms

35. First Shifting Theorem
If Lff (t)g = f (s) then L
eatf (t)
= f (s a)
N. B. Vyas Laplace Transforms

36. First Shifting Theorem
If Lff (t)g = f (s) then L
eatf (t)
= f (s a)
Proof: By the def. of Laplace
N. B. Vyas Laplace Transforms

37. First Shifting Theorem
If Lff (t)g = f (s) then L
eatf (t)
= f (s a)
Proof: By the def. of Laplace
1R
L
eatf (t)
=
0
esteatf (t) dt
N. B. Vyas Laplace Transforms

38. First Shifting Theorem
If Lff (t)g = f (s) then L
eatf (t)
= f (s a)
Proof: By the def. of Laplace
1R
L
eatf (t)
=
0
esteatf (t) dt
=
1R
0
e(sa)tf (t) dt
N. B. Vyas Laplace Transforms

39. First Shifting Theorem
If Lff (t)g = f (s) then L
eatf (t)
= f (s a)
Proof: By the def. of Laplace
1R
L
eatf (t)
=
0
esteatf (t) dt
=
1R
0
e(sa)tf (t) dt
=
1R
0
ertf (t) dt
N. B. Vyas Laplace Transforms

40. First Shifting Theorem
If Lff (t)g = f (s) then L
eatf (t)
= f (s a)
Proof: By the def. of Laplace
1R
L
eatf (t)
=
0
esteatf (t) dt
=
1R
0
e(sa)tf (t) dt
=
1R
0
ertf (t) dt
= f(r)
N. B. Vyas Laplace Transforms

41. First Shifting Theorem
If Lff (t)g = f (s) then L
eatf (t)
= f (s a)
Proof: By the def. of Laplace
1R
L
eatf (t)
=
0
esteatf (t) dt
=
1R
0
e(sa)tf (t) dt
=
1R
0
ertf (t) dt
= f(r)
= f(s a)
N. B. Vyas Laplace Transforms

42. First Shifting Theorem
If Lff (t)g = f (s) then L
eatf (t)
= f (s a)
Proof: By the def. of Laplace
1R
L
eatf (t)
=
0
esteatf (t) dt
=
1R
0
e(sa)tf (t) dt
=
1R
0
ertf (t) dt
= f(r)
= f(s a)
Thus if we know the transformation f(s) of f(t) then we can
write the transformation of eatf(t) simply replacing s by s a to
get F(s a)
N. B. Vyas Laplace Transforms

43. First Shifting Theorem
Note:
1 L(eat) =
1
s a
N. B. Vyas Laplace Transforms

44. First Shifting Theorem
Note:
1 L(eat) =
1
s a
2 L[eattn] =
(n + 1)
(s a)n+1
N. B. Vyas Laplace Transforms

45. First Shifting Theorem
Note:
1 L(eat) =
1
s a
2 L[eattn] =
(n + 1)
(s a)n+1
3 L[eatsinh bt] =
b
(s a)2 b2
N. B. Vyas Laplace Transforms

46. First Shifting Theorem
Note:
1 L(eat) =
1
s a
2 L[eattn] =
(n + 1)
(s a)n+1
3 L[eatsinh bt] =
b
(s a)2 b2
4 L[eatcosh bt] =
s
(s a)2 b2
N. B. Vyas Laplace Transforms

47. First Shifting Theorem
Note:
1 L(eat) =
1
s a
2 L[eattn] =
(n + 1)
(s a)n+1
3 L[eatsinh bt] =
b
(s a)2 b2
4 L[eatcosh bt] =
s
(s a)2 b2
5 L[eatsin bt] =
b
(s a)2 + b2
N. B. Vyas Laplace Transforms

48. First Shifting Theorem
Note:
1 L(eat) =
1
s a
2 L[eattn] =
(n + 1)
(s a)n+1
3 L[eatsinh bt] =
b
(s a)2 b2
4 L[eatcosh bt] =
s
(s a)2 b2
5 L[eatsin bt] =
b
(s a)2 + b2
6 L[eatcos bt] =
s a
(s a)2 + b2 ; s 0
N. B. Vyas Laplace Transforms

49. Examples
1 Find out the Laplace transform of e3t (2 cos 5t 3 sin 5t)
N. B. Vyas Laplace Transforms

50. Examples
1 Find out the Laplace transform of e3t (2 cos 5t 3 sin 5t)
2 L[eatsinhbt]
N. B. Vyas Laplace Transforms

51. Examples
1 Find out the Laplace transform of e3t (2 cos 5t 3 sin 5t)
2 L[eatsinhbt]
3 L[t3e3t]
N. B. Vyas Laplace Transforms

52. Examples
1 Find out the Laplace transform of e3t (2 cos 5t 3 sin 5t)
2 L[eatsinhbt]
3 L[t3e3t]
4 L[(t + 2)2et]
N. B. Vyas Laplace Transforms

53. Examples
1 Find out the Laplace transform of e3t (2 cos 5t 3 sin 5t)
2 L[eatsinhbt]
3 L[t3e3t]
4 L[(t + 2)2et]
5 L[etsin2t]
N. B. Vyas Laplace Transforms

54. Examples
1 Find out the Laplace transform of e3t (2 cos 5t 3 sin 5t)
2 L[eatsinhbt]
3 L[t3e3t]
4 L[(t + 2)2et]
5 L[etsin2t]
6 L[cosh at sin at]
N. B. Vyas Laplace Transforms

55. Examples
Ex. Find the Laplace transform of the function which is de

56. ned as
f(t) =
t/T 0 t T
1 when t T
N. B. Vyas Laplace Transforms

57. Examples
Ex. Find Laplace transform of f(t) =
sin t 0 t
0 when t
N. B. Vyas Laplace Transforms

58. Examples
Ex. Find Laplace transform of f(t) where f(t) =
t 0 t 4
5 when t 4
N. B. Vyas Laplace Transforms

59. Change of Scale property
If Lff (t)g = f (s) then L
eatf (bt)
=
1
b
f
s a
b
; b 0
N. B. Vyas Laplace Transforms

60. Change of Scale property
If Lff (t)g = f (s) then L
eatf (bt)
=
1
b
f
s a
b
; b 0
Proof: By the def. of Laplace
N. B. Vyas Laplace Transforms

61. Change of Scale property
If Lff (t)g = f (s) then L
eatf (bt)
=
1
b
f
s a
b
; b 0
Proof: By the def. of Laplace
Lff (t)g =
1 Z
0
estf (t) dt
N. B. Vyas Laplace Transforms

62. Change of Scale property
If Lff (t)g = f (s) then L
eatf (bt)
=
1
b
f
s a
b
; b 0
Proof: By the def. of Laplace
Lff (t)g =
1 Z
0
estf (t) dt
L
eatf (bt)
=
1 Z
0
esteatf (bt) dt
N. B. Vyas Laplace Transforms

63. Change of Scale property
If Lff (t)g = f (s) then L
eatf (bt)
=
1
b
f
s a
b
; b 0
Proof: By the def. of Laplace
Lff (t)g =
1 Z
0
estf (t) dt
L
eatf (bt)
=
1 Z
0
esteatf (bt) dt
=
1R
0
e(sa)tf (bt) dt
N. B. Vyas Laplace Transforms

64. Change of Scale property
If Lff (t)g = f (s) then L
eatf (bt)
=
1
b
f
s a
b
; b 0
Proof: By the def. of Laplace
Lff (t)g =
1 Z
0
estf (t) dt
L
eatf (bt)
=
1 Z
0
esteatf (bt) dt
=
1R
0
e(sa)tf (bt) dt
=
1R
0
e( sa
b )btf (bt) dt
N. B. Vyas Laplace Transforms

65. Change of Scale property
If Lff (t)g = f (s) then L
eatf (bt)
=
1
b
f
s a
b
; b 0
Proof: By the def. of Laplace
Lff (t)g =
1 Z
0
estf (t) dt
L
eatf (bt)
=
1 Z
0
esteatf (bt) dt
=
1R
0
e(sa)tf (bt) dt
=
1R
0
e( sa
b )btf (bt) dt
Let bt = u ) b dt = du
N. B. Vyas Laplace Transforms

66. Change of Scale property
If Lff (t)g = f (s) then L
eatf (bt)
=
1
b
f
s a
b
; b 0
Proof: By the def. of Laplace
Lff (t)g =
1 Z
0
estf (t) dt
L
eatf (bt)
=
1 Z
0
esteatf (bt) dt
=
1R
0
e(sa)tf (bt) dt
=
1R
0
e( sa
b )btf (bt) dt
Let bt = u ) b dt = du
1R
L
eatf (bt)
=
0
e( sa
b )uf (u)
du
b
N. B. Vyas Laplace Transforms

67. Change of Scale property
If Lff (t)g = f (s) then L
eatf (bt)
=
1
b
f
s a
b
; b 0
Proof: By the def. of Laplace
Lff (t)g =
1 Z
0
estf (t) dt
L
eatf (bt)
=
1 Z
0
esteatf (bt) dt
=
1R
0
e(sa)tf (bt) dt
=
1R
0
e( sa
b )btf (bt) dt
Let bt = u ) b dt = du
1R
L
eatf (bt)
=
0
e( sa
b )uf (u)
du
b
=
1
b
f
s a
b
N. B. Vyas Laplace Transforms

68. Inverse Laplace Transform
If Lff (t)g = f (s) then f(t) is called the inverse
Laplace
transform of f(s) and it is denoted by L1
f(s)
= f (t)
N. B. Vyas Laplace Transforms

69. Inverse Laplace Transform
If Lff (t)g = f (s) then f(t) is called the inverse
Laplace
transform of f(s) and it is denoted by L1
f(s)
= f (t)
1 L1
1
s
= 1
N. B. Vyas Laplace Transforms

70. Inverse Laplace Transform
If Lff (t)g = f (s) then f(t) is called the inverse
Laplace
transform of f(s) and it is denoted by L1
f(s)
= f (t)
1 L1
1
s
= 1
2 L1
1
s a
= eat
N. B. Vyas Laplace Transforms

71. Inverse Laplace Transform
If Lff (t)g = f (s) then f(t) is called the inverse
Laplace
transform of f(s) and it is denoted by L1
f(s)
= f (t)
1 L1
1
s
= 1
2 L1
1
s a
= eat
3 L1
1
s2 + a2
=
1
a
sin at
N. B. Vyas Laplace Transforms

72. Inverse Laplace Transform
If Lff (t)g = f (s) then f(t) is called the inverse
Laplace
transform of f(s) and it is denoted by L1
f(s)
= f (t)
1 L1
1
s
= 1
2 L1
1
s a
= eat
3 L1
1
s2 + a2
=
1
a
sin at
4 L1
s
s2 + a2
= cos at
N. B. Vyas Laplace Transforms

73. Inverse Laplace Transform
If Lff (t)g = f (s) then f(t) is called the inverse
Laplace
transform of f(s) and it is denoted by L1
f(s)
= f (t)
1 L1
1
s
= 1
2 L1
1
s a
= eat
3 L1
1
s2 + a2
=
1
a
sin at
4 L1
s
s2 + a2
= cos at
5 L1
1
s2 a2
=
1
a
sinh at
N. B. Vyas Laplace Transforms

74. Inverse Laplace Transform
If Lff (t)g = f (s) then f(t) is called the inverse
Laplace
transform of f(s) and it is denoted by L1
f(s)
= f (t)
1 L1
1
s
= 1
2 L1
1
s a
= eat
3 L1
1
s2 + a2
=
1
a
sin at
4 L1
s
s2 + a2
= cos at
5 L1
1
s2 a2
=
1
a
sinh at
6 L1
s
s2 a2
= cosh at
N. B. Vyas Laplace Transforms

75. Inverse Laplace Transform
If Lff (t)g = f (s) then f(t) is called the inverse
Laplace
transform of f(s) and it is denoted by L1
f(s)
= f (t)
1 L1
1
s
= 1
2 L1
1
s a
= eat
3 L1
1
s2 + a2
=
1
a
sin at
4 L1
s
s2 + a2
= cos at
5 L1
1
s2 a2
=
1
a
sinh at
6 L1
s
s2 a2
= cosh at
7 L1
1
sn
=
tn1
(n 1)!
N. B. Vyas Laplace Transforms

76. Partial Fractions
N. B. Vyas Laplace Transforms

77. Partial Fractions
N. B. Vyas Laplace Transforms

78. Partial Fractions
N. B. Vyas Laplace Transforms

79. Examples of Inverse Laplace Transform - I
1 L1
s2 3s + 4
s3
N. B. Vyas Laplace Transforms

80. Examples of Inverse Laplace Transform - I
1 L1
s2 3s + 4
s3
2 L1
3
2
s4 2s2 + 1
s5
N. B. Vyas Laplace Transforms

81. Examples of Inverse Laplace Transform - I
1 L1
s2 3s + 4
s3
2 L1
3
2
s4 2s2 + 1
s5
3 L1
s + 7
(s + 1)2 + 1
N. B. Vyas Laplace Transforms

82. Examples of Inverse Laplace Transform - I
1 L1
s2 3s + 4
s3
2 L1
3
2
s4 2s2 + 1
s5
3 L1
s + 7
(s + 1)2 + 1
4 L1
3s + 5
(s + 1)4
N. B. Vyas Laplace Transforms

83. Examples of Inverse Laplace Transform - I
1 L1
s2 3s + 4
s3
2 L1
3
2
s4 2s2 + 1
s5
3 L1
s + 7
(s + 1)2 + 1
4 L1
3s + 5
(s + 1)4
5 L1
3s
s2 + 2s 8
N. B. Vyas Laplace Transforms

84. Examples of Inverse Laplace Transform - I
1 L1
s2 3s + 4
s3
2 L1
3
2
s4 2s2 + 1
s5
3 L1
s + 7
(s + 1)2 + 1
4 L1
3s + 5
(s + 1)4
5 L1
3s
s2 + 2s 8
6 L1
3s + 7
s2 2s 3
N. B. Vyas Laplace Transforms

85. Examples of Inverse Laplace Transform - I
1 L1
s2 3s + 4
s3
2 L1
3
2
s4 2s2 + 1
s5
3 L1
s + 7
(s + 1)2 + 1
4 L1
3s + 5
(s + 1)4
5 L1
3s
s2 + 2s 8
6 L1
3s + 7
s2 2s 3
7 L1
2s2 6s + 5
s3 6s2 + 11s 6
N. B. Vyas Laplace Transforms

86. Examples of Inverse Laplace Transform - II
1 L1
s + 29
(s + 4)(s2 + 9)
N. B. Vyas Laplace Transforms

87. Examples of Inverse Laplace Transform - II
1 L1
s + 29
(s + 4)(s2 + 9)
2 L1
s
(s2 1)
N. B. Vyas Laplace Transforms

88. Examples of Inverse Laplace Transform - II
1 L1
s + 29
(s + 4)(s2 + 9)
2 L1
s
(s2 1)
3 L1
4s + 5
(s 1)2(s + 2)
N. B. Vyas Laplace Transforms

89. Examples of Inverse Laplace Transform - II
1 L1
s + 29
(s + 4)(s2 + 9)
2 L1
s
(s2 1)
3 L1
4s + 5
(s 1)2(s + 2)
4 L1
2s2 1
(s2 + 1)(s2 + 4)
N. B. Vyas Laplace Transforms

90. Examples of Inverse Laplace Transform - II
1 L1
s + 29
(s + 4)(s2 + 9)
2 L1
s
(s2 1)
3 L1
4s + 5
(s 1)2(s + 2)
4 L1
2s2 1
(s2 + 1)(s2 + 4)
5 L1
s
s4 + s2 + 1
N. B. Vyas Laplace Transforms

91. Examples of Inverse Laplace Transform - II
1 L1
s + 29
(s + 4)(s2 + 9)
2 L1
s
(s2 1)
3 L1
4s + 5
(s 1)2(s + 2)
4 L1
2s2 1
(s2 + 1)(s2 + 4)
5 L1
s
s4 + s2 + 1
6 L1
s
s4 + 4a4
N. B. Vyas Laplace Transforms

92. Examples of Inverse Laplace Transform - II
1 L1
s + 29
(s + 4)(s2 + 9)
2 L1
s
(s2 1)
3 L1
4s + 5
(s 1)2(s + 2)
4 L1
2s2 1
(s2 + 1)(s2 + 4)
5 L1
s
s4 + s2 + 1
6 L1
s
s4 + 4a4
7 L1
s + 3
s2 + 6s + 13
N. B. Vyas Laplace Transforms

93. Transformation of Derivatives
Thm: If f0(t) be continuous and L[f(t)] = f(s) then
Lff0(t)g = s f(s) f(0) provided lim
t!1
estf(t) = 0
N. B. Vyas Laplace Transforms

94. Transformation of Derivatives
Thm: If f0(t) be continuous and L[f(t)] = f(s) then
Lff0(t)g = s f(s) f(0) provided lim
t!1
estf(t) = 0
i.e. Lff0(t)g = sL ff(t)g f(0)
N. B. Vyas Laplace Transforms

95. Transformation of Derivatives
Thm: If f0(t) be continuous and L[f(t)] = f(s) then
Lff0(t)g = s f(s) f(0) provided lim
t!1
estf(t) = 0
i.e. Lff0(t)g = sL ff(t)g f(0)
Simillarly Lff00(t)g = sL ff0(t)g f0(0)
N. B. Vyas Laplace Transforms

96. Transformation of Derivatives
Thm: If f0(t) be continuous and L[f(t)] = f(s) then
Lff0(t)g = s f(s) f(0) provided lim
t!1
estf(t) = 0
i.e. Lff0(t)g = sL ff(t)g f(0)
Simillarly Lff00(t)g = sL ff0(t)g f0(0)
= s [sL ff(t)g f(0)] f0(0)
N. B. Vyas Laplace Transforms

97. Transformation of Derivatives
Thm: If f0(t) be continuous and L[f(t)] = f(s) then
Lff0(t)g = s f(s) f(0) provided lim
t!1
estf(t) = 0
i.e. Lff0(t)g = sL ff(t)g f(0)
Simillarly Lff00(t)g = sL ff0(t)g f0(0)
= s [sL ff(t)g f(0)] f0(0)
= s2Lff(t)g sf(0) f0(0)
N. B. Vyas Laplace Transforms

98. Transformation of Derivatives
Thm: If f0(t) be continuous and L[f(t)] = f(s) then
Lff0(t)g = s f(s) f(0) provided lim
t!1
estf(t) = 0
i.e. Lff0(t)g = sL ff(t)g f(0)
Simillarly Lff00(t)g = sL ff0(t)g f0(0)
= s [sL ff(t)g f(0)] f0(0)
= s2Lff(t)g sf(0) f0(0)
= s2 f(s) sf(0) f0(0)
N. B. Vyas Laplace Transforms

99. Transformation of Derivatives
Thm: If f0(t) be continuous and L[f(t)] = f(s) then
Lff0(t)g = s f(s) f(0) provided lim
t!1
estf(t) = 0
i.e. Lff0(t)g = sL ff(t)g f(0)
Simillarly Lff00(t)g = sL ff0(t)g f0(0)
= s [sL ff(t)g f(0)] f0(0)
= s2Lff(t)g sf(0) f0(0)
= s2 f(s) sf(0) f0(0)
In general
Lffn(t)g = sn f(s) sn1f(0) sn2f0(0) : : : fn1(0)
N. B. Vyas Laplace Transforms

100. Transformation of Derivatives
Thm: If f0(t) be continuous and L[f(t)] = f(s) then
Lff0(t)g = s f(s) f(0) provided lim
t!1
estf(t) = 0
i.e. Lff0(t)g = sL ff(t)g f(0)
Simillarly Lff00(t)g = sL ff0(t)g f0(0)
= s [sL ff(t)g f(0)] f0(0)
= s2Lff(t)g sf(0) f0(0)
= s2 f(s) sf(0) f0(0)
In general
Lffn(t)g = sn f(s) sn1f(0) sn2f0(0) : : : fn1(0)
N. B. Vyas Laplace Transforms

101. Examples of Transformation of Derivatives
Ex. Derive the Laplace transform of sin at and cos at
N. B. Vyas Laplace Transforms

102. Examples of Transformation of Derivatives
Ex. Derive the Laplace transform of sin at and cos at
Ex. Obtain Lftng from L(1) =
1
s
N. B. Vyas Laplace Transforms

103. Examples of Transformation of Derivatives
Ex. Derive the Laplace transform of sin at and cos at
Ex. Obtain Lftng from L(1) =
1
s
Ex. Find L(t sin at)
N. B. Vyas Laplace Transforms

104. Examples of Transformation of Derivatives
Ex. Derive the Laplace transform of sin at and cos at
Ex. Obtain Lftng from L(1) =
1
s
Ex. Find L(t sin at)
Ex. Find L(t cos at)
N. B. Vyas Laplace Transforms

105. Transformation of Integrals
Thm: If L[f(t)] = f(s) then L
Z t
0
f(u)du
=
1
s
f(s)
N. B. Vyas Laplace Transforms

106. Transformation of Integrals
Thm: If L[f(t)] = f(s) then L
Z t
0
f(u)du
=
1
s
f(s)
Proof: Let I(t) =
Z t
0
f(u)du
N. B. Vyas Laplace Transforms

107. Transformation of Integrals
Thm: If L[f(t)] = f(s) then L
Z t
0
f(u)du
=
1
s
f(s)
Proof: Let I(t) =
Z t
0
f(u)du
) I0(t) =
d
dt
Z t
0
= f(t)
f(u)du
N. B. Vyas Laplace Transforms

108. Transformation of Integrals
Thm: If L[f(t)] = f(s) then L
Z t
0
f(u)du
=
1
s
f(s)
Proof: Let I(t) =
Z t
0
f(u)du
) I0(t) =
d
dt
Z t
0
= f(t) and I(0) = 0
f(u)du
N. B. Vyas Laplace Transforms

109. Transformation of Integrals
Thm: If L[f(t)] = f(s) then L
Z t
0
f(u)du
=
1
s
f(s)
Proof: Let I(t) =
Z t
0
f(u)du
) I0(t) =
d
dt
Z t
0
= f(t) and I(0) = 0
f(u)du
) Lff(t)g = LfI0(t)g = sI(s) I(0) = sI(s)
N. B. Vyas Laplace Transforms

110. Transformation of Integrals
Thm: If L[f(t)] = f(s) then L
Z t
0
f(u)du
=
1
s
f(s)
Proof: Let I(t) =
Z t
0
f(u)du
) I0(t) =
d
dt
Z t
0
= f(t) and I(0) = 0
f(u)du
) Lff(t)g = LfI0(t)g = sI(s) I(0) = sI(s)
) Lff(t)g = sI(s)
N. B. Vyas Laplace Transforms

111. Transformation of Integrals
Thm: If L[f(t)] = f(s) then L
Z t
0
f(u)du
=
1
s
f(s)
Proof: Let I(t) =
Z t
0
f(u)du
) I0(t) =
d
dt
Z t
0
= f(t) and I(0) = 0
f(u)du
) Lff(t)g = LfI0(t)g = sI(s) I(0) = sI(s)
) Lff(t)g = sI(s)
) Lff(t)g = sL fI(t)g
N. B. Vyas Laplace Transforms

112. Transformation of Integrals
Thm: If L[f(t)] = f(s) then L
Z t
0
f(u)du
=
1
s
f(s)
Proof: Let I(t) =
Z t
0
f(u)du
) I0(t) =
d
dt
Z t
0
= f(t) and I(0) = 0
f(u)du
) Lff(t)g = LfI0(t)g = sI(s) I(0) = sI(s)
) Lff(t)g = sI(s)
) Lff(t)g = Z sL fI(t)g
t
) f(s) = sL
0
f(u)du
N. B. Vyas Laplace Transforms

113. Transformation of Integrals
Thm: If L[f(t)] = f(s) then L
Z t
0
f(u)du
=
1
s
f(s)
Proof: Let I(t) =
Z t
0
f(u)du
) I0(t) =
d
dt
Z t
0
= f(t) and I(0) = 0
f(u)du
) Lff(t)g = LfI0(t)g = sI(s) I(0) = sI(s)
) Lff(t)g = sI(s)
) Lff(t)g = Z sL fI(t)g
t
) f(s) = sL
0
f(u)du
) 1
s
f(s) = L
Z t
0
f(u)du
N. B. Vyas Laplace Transforms

114. Transformation of Integrals
Thm: If L[f(t)] = f(s) then L
Z t
0
f(u)du
=
1
s
f(s)
Proof: Let I(t) =
Z t
0
f(u)du
) I0(t) =
d
dt
Z t
0
= f(t) and I(0) = 0
f(u)du
) Lff(t)g = LfI0(t)g = sI(s) I(0) = sI(s)
) Lff(t)g = sI(s)
) Lff(t)g = Z sL fI(t)g
t
) f(s) = sL
0
f(u)du
) 1
s
f(s) = L
Z t
0
f(u)du
) L1
1
s
f(s)
=
Z t
0
f(u)du
N. B. Vyas Laplace Transforms

115. Examples of Transformation of Integrals
Ex. Prove that: L1
1
s2 + 1
= sin t
Ex. Prove that: L1
1
s(s2 + 1)
= 1 cos t
Ex. Find inverse Laplace transform of
1
s3(s2 + a2)
N. B. Vyas Laplace Transforms

116. Multiplication by tn
Thm: If L[f(t)] = f(s) then Lftnf(t)g = (1)n dn
dsn
f(s)
N. B. Vyas Laplace Transforms

117. Multiplication by tn
Thm: If L[f(t)] = f(s) then Lftnf(t)g = (1)n dn
dsn
f(s)
if Lftf (t)g = (1)1 d
ds
f(s)
then L1
f0(s)
= tf (t)
N. B. Vyas Laplace Transforms

118. Examples of Laplace transform when tn is in
multiplication
1 L
t2eat
N. B. Vyas Laplace Transforms

119. Examples of Laplace transform when tn is in
multiplication
1 L
t2eat
2 L
t3e3t
N. B. Vyas Laplace Transforms

120. Examples of Laplace transform when tn is in
multiplication
1 L
t2eat
2 L
t3e3t
3 Lftcos atg
N. B. Vyas Laplace Transforms

121. Examples of Laplace transform when tn is in
multiplication
1 L
t2eat
2 L
t3e3t
3 Lftcos atg
4 L
tsin2t
N. B. Vyas Laplace Transforms

122. Examples of Laplace transform when tn is in
multiplication
1 L
t2eat
2 L
t3e3t
3 Lftcos atg
4 L
tsin2t
5 L
te2tcos 3t
N. B. Vyas Laplace Transforms

123. Examples of Laplace transform when tn is in
multiplication
1 L
t2eat
2 L
t3e3t
3 Lftcos atg
4 L
tsin2t
5 L
te2tcos 3t
6 Lftcos(4t + 3)g
N. B. Vyas Laplace Transforms

124. Division by t
Thm: If L[f(t)] = f(s) then L
1
t
f(t)
=
Z 1
s
f(s) provided the
integral exists.
N. B. Vyas Laplace Transforms

125. Examples of Laplace Transform when t is in division
1 L
sin t
t
N. B. Vyas Laplace Transforms

126. Examples of Laplace Transform when t is in division
1 L
sin t
t
2 L
1 cos 2t
t
N. B. Vyas Laplace Transforms

127. Examples of Laplace Transform when t is in division
1 L
sin t
t
2 L
1 cos 2t
t
3 L
eat ebt
t
N. B. Vyas Laplace Transforms

128. Examples of Laplace Transform when t is in division
1 L
sin t
t
2 L
1 cos 2t
t
3 L
eat ebt
t
4 L
cos 2t cos 3t
t
N. B. Vyas Laplace Transforms

129. Examples of Laplace Transform when t is in division
1 L
sin t
t
2 L
1 cos 2t
t
3 L
eat ebt
t
4 L
cos 2t cos 3t
t
5 L
1 et
t
N. B. Vyas Laplace Transforms

130. Examples of Laplace Transform when t is in division
1 L
sin t
t
2 L
1 cos 2t
t
3 L
eat ebt
t
4 L
cos 2t cos 3t
t
5 L
1 et
t
6 L
cos at cos bt
t
N. B. Vyas Laplace Transforms

131. Examples of Laplace Transform when t is in division
1 L
sin t
t
2 L
1 cos 2t
t
3 L
eat ebt
t
4 L
cos 2t cos 3t
t
5 L
1 et
t
6 L
cos at cos bt
t
7 L
etsin t
t
N. B. Vyas Laplace Transforms

132. Examples of in

133. nite integral using Laplace Transform
1 Find
Z 1
0
te2tsin t dt
N. B. Vyas Laplace Transforms

134. Examples of in

135. nite integral using Laplace Transform
1 Find
Z 1
0
te2tsin t dt
2 Find
Z 1
0
sin mt
t
dt
N. B. Vyas Laplace Transforms

136. Examples of in

137. nite integral using Laplace Transform
1 Find
Z 1
0
te2tsin t dt
2 Find
Z 1
0
sin mt
t
dt
3 Find
Z 1
0
et e3t
t
dt
N. B. Vyas Laplace Transforms

138. Examples of in

139. nite integral using Laplace Transform
1 Find
Z 1
0
te2tsin t dt
2 Find
Z 1
0
sin mt
t
dt
3 Find
Z 1
0
et e3t
t
dt
4 Find
Z 1
0
etsin2 t
t
dt
N. B. Vyas Laplace Transforms

140. Examples of Inverse Laplace Transform
1 L1
s
(s2 + a2)2
N. B. Vyas Laplace Transforms

141. Examples of Inverse Laplace Transform
1 L1
s
(s2 + a2)2
2 L1
cot1 s
a
N. B. Vyas Laplace Transforms

142. Examples of Inverse Laplace Transform
1 L1
s
(s2 + a2)2
2 L1
cot1 s
a
3 L1
log
s + 1
s 1
N. B. Vyas Laplace Transforms

143. Convolution
Defn:
Convolution of function Z f(t) and g(t) is denoted f(t) g(t) and
t
de

144. ned as f(t) g(t) =
0
f(u)g(t u) du
N. B. Vyas Laplace Transforms

145. Convolution
Defn:
Convolution of function Z f(t) and g(t) is denoted f(t) g(t) and
t
de

146. ned as f(t) g(t) =
0
f(u)g(t u) du
Theorem:
Convolution theorem
If L1
f(s)
= f(t) and L1 fg(s)g = g(t) then
L1
f(s)g(s)
=
Z t
0
f(u)g(t u) du
N. B. Vyas Laplace Transforms

147. Convolution
Proof: Let (t) =
Z t
0
f(u)g(t u) du
N. B. Vyas Laplace Transforms

148. Convolution
Proof: Let (t) =
Z t
0
f(u)g(t u) du
then L((t)) =
Z 1
0
est
Z t
0
dt
f(u)g(t u) du
N. B. Vyas Laplace Transforms

149. Convolution
Proof: Let (t) =
Z t
0
f(u)g(t u) du
then L((t)) =
Z 1
0
est
Z t
0
dt
f(u)g(t u) du
=
Z 1
0
Z t
0
estf(u)g(t u) du dt
N. B. Vyas Laplace Transforms

150. Convolution
Proof: Let (t) =
Z t
0
f(u)g(t u) du
then L((t)) =
Z 1
0
est
Z t
0
dt
f(u)g(t u) du
=
Z 1
0
Z t
0
estf(u)g(t u) du dt
The region integration for this double integration is entire area
lying between the lines u = 0 and u = t. On changing the order
of integration, we get
N. B. Vyas Laplace Transforms

151. Convolution
Proof: Let (t) =
Z t
0
f(u)g(t u) du
then L((t)) =
Z 1
0
est
Z t
0
dt
f(u)g(t u) du
=
Z 1
0
Z t
0
estf(u)g(t u) du dt
The region integration for this double integration is entire area
lying between the lines u = 0 and u = t. On changing the order
of integration, Z we get
1
L((t)) =
0
Z 1
u
estf(u)g(t u) dt du
N. B. Vyas Laplace Transforms

152. Convolution
Proof: Let (t) =
Z t
0
f(u)g(t u) du
then L((t)) =
Z 1
0
est
Z t
0
dt
f(u)g(t u) du
=
Z 1
0
Z t
0
estf(u)g(t u) du dt
The region integration for this double integration is entire area
lying between the lines u = 0 and u = t. On changing the order
of integration, Z we get
1
L((t)) =
0
Z 1
u
estf(u)g(t u) dt du
=
Z 1
0
esuf(u)
Z 1
u
du
est+sug(t u) dt
N. B. Vyas Laplace Transforms

153. Convolution
Proof: Let (t) =
Z t
0
f(u)g(t u) du
then L((t)) =
Z 1
0
est
Z t
0
dt
f(u)g(t u) du
=
Z 1
0
Z t
0
estf(u)g(t u) du dt
The region integration for this double integration is entire area
lying between the lines u = 0 and u = t. On changing the order
of integration, Z we get
1
L((t)) =
0
Z 1
u
estf(u)g(t u) dt du
=
Z 1
0
esuf(u)
Z 1
u
du
est+sug(t u) dt
=
Z 1
0
esuf(u)
Z 1
u
du
es(tu)g(t u) dt
N. B. Vyas Laplace Transforms

154. Convolution
Proof: Let (t) =
Z t
0
f(u)g(t u) du
then L((t)) =
Z 1
0
est
Z t
0
dt
f(u)g(t u) du
=
Z 1
0
Z t
0
estf(u)g(t u) du dt
The region integration for this double integration is entire area
lying between the lines u = 0 and u = t. On changing the order
of integration, Z we get
1
L((t)) =
0
Z 1
u
estf(u)g(t u) dt du
=
Z 1
0
esuf(u)
Z 1
u
du
est+sug(t u) dt
=
Z 1
0
esuf(u)
Z 1
u
du
es(tu)g(t u) dt
=
Z 1
0
esuf(u)
Z 1
u
esvg(v) dv
du, Putting t u = v
N. B. Vyas Laplace Transforms

155. Convolution
=
Z 1
0
esuf(u)g(s)du
N. B. Vyas Laplace Transforms

156. Convolution
=
Z 1
0
esuf(u)g(s)du
= g(s)
Z 1
0
esuf(u)du
N. B. Vyas Laplace Transforms

157. Convolution
=
Z 1
0
esuf(u)g(s)du
= g(s)
Z 1
0
esuf(u)du
) L((t)) = g(s) f(s)
N. B. Vyas Laplace Transforms

158. Convolution
=
Z 1
0
esuf(u)g(s)du
= g(s)
Z 1
0
esuf(u)du
) L((t)) = g(s) f(s)
L1
g(s) f(s)
= (t) =
Z t
0
f(u)g(t u) du
N. B. Vyas Laplace Transforms

159. Examples of Convolution theorem
Apply convolution theorem to evaluate:
Ex. L1
1
s2(s 1)
N. B. Vyas Laplace Transforms

160. Examples of Convolution theorem
Apply convolution theorem to evaluate:
Ex. L1
1
s2(s 1)
Ex. L1
s
(s2 + 4)2
N. B. Vyas Laplace Transforms

161. Examples of Convolution theorem
Apply convolution theorem to evaluate:
Ex. L1
1
s2(s 1)
Ex. L1
s
(s2 + 4)2
Ex. L1
1
(s + a)(s + b)
N. B. Vyas Laplace Transforms

162. Examples of Convolution theorem
Apply convolution theorem to evaluate:
Ex. L1
1
s2(s 1)
Ex. L1
s
(s2 + 4)2
Ex. L1
1
(s + a)(s + b)
Ex. L1
1
s(s2 + 4)
N. B. Vyas Laplace Transforms

163. Application to Dierential Equations
Ex. Use transform method to solve y00 + 3y0 + 2y = et, y(0) = 1 ,
y0(0) = 0
N. B. Vyas Laplace Transforms

164. Application to Dierential Equations
Ex. Use transform method to solve y00 + 3y0 + 2y = et, y(0) = 1 ,
y0(0) = 0
Ex. Solve the equation x00 + 2x0 + 5x = et sin t, x(0) = 0 , x0(0) = 1
N. B. Vyas Laplace Transforms

165. Laplace transform of Periodic function
If f(t) is sectionally continuous function over an
interval of length p (0 t p) and f(t) is a
periodic function with period p (p 0), that is
f(t + p) = f(t), then its Laplace transform exists
and
1
Lff(t)g =
1 eps
Z p
0
estf(t)dt, (s 0)
N. B. Vyas Laplace Transforms

166. Laplace transform of Periodic function
Periodic Square Wave
Ex. Find the Laplace transform of the square wave
function of period 2a de

167. ned as
f(t) =
k if 0 t a
k if a t 2a
N. B. Vyas Laplace Transforms

168. Laplace transform of Periodic function
Periodic Triangular Wave
Ex. Find the Laplace transform of periodic function
f(t) =
t if 0 t a
2a t if a t 2a
with period 2a
N. B. Vyas Laplace Transforms

169. Unit Step function or Heaviside's unit function
The Heaviside step function, or the unit step
function, usually denoted by H (but sometimes u
or ), is a discontinuous function whose value is
zero for negative argument and one for positive
argument.
N. B. Vyas Laplace Transforms

170. Unit Step function or Heaviside's unit function
The Heaviside step function, or the unit step
function, usually denoted by H (but sometimes u
or ), is a discontinuous function whose value is
zero for negative argument and one for positive
argument.
The function is used in the mathematics of control
theory, signal processing, structural mechanics,
etc..
N. B. Vyas Laplace Transforms

171. Unit Step function or Heaviside's unit function
It is denoted by ua(t) or
u(t a) or H(t a) and
is de

172. ned as H(t a) =
0 t a
1 t a
N. B. Vyas Laplace Transforms