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.
ENGLISH 7_Q4_LESSON 2_ Employing a Variety of Strategies for Effective Interp...
Laplace Transforms Explained
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
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
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
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
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
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
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
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
173. 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
174. ned as H(t a) =
0 t a
1 t a
In particular,
when a = 0
H(t) =
0 t 0
1 t 0
N. B. Vyas Laplace Transforms
175. Unit Step function or Heaviside's unit function
N. B. Vyas Laplace Transforms
176. Unit Step function or Heaviside's unit function
Laplace Transform of Unit Step Function:
By de
177. nition of Z Laplace transform
1
Lfu(t a)g =
0
estu(t a)dt
N. B. Vyas Laplace Transforms
178. Unit Step function or Heaviside's unit function
Laplace Transform of Unit Step Function:
By de
179. nition of Z Laplace transform
1
Lfu(t a)g =
0
estu(t a)dt
=
Z a
0
est(0)dt +
Z 1
a
est(1)dt
N. B. Vyas Laplace Transforms
180. Unit Step function or Heaviside's unit function
Laplace Transform of Unit Step Function:
By de
181. nition of Z Laplace transform
1
Lfu(t a)g =
0
estu(t a)dt
=
Z a
0
est(0)dt +
Z 1
a
est(1)dt
=
Z 1
a
estdt =
est
s
1
a
=
1
s
eas
N. B. Vyas Laplace Transforms
182. Unit Step function or Heaviside's unit function
Laplace Transform of Unit Step Function:
By de
183. nition of Z Laplace transform
1
Lfu(t a)g =
0
estu(t a)dt
=
Z a
0
est(0)dt +
Z 1
a
est(1)dt
=
Z 1
a
estdt =
est
s
1
a
=
1
s
eas
) L1
1
s
eas
= u(t a)
N. B. Vyas Laplace Transforms
184. Unit Step function or Heaviside's unit function
Laplace Transform of Unit Step Function:
By de
185. nition of Z Laplace transform
1
Lfu(t a)g =
0
estu(t a)dt
=
Z a
0
est(0)dt +
Z 1
a
est(1)dt
=
Z 1
a
estdt =
est
s
1
a
=
1
s
eas
) L1
1
s
eas
= u(t a)
In particular, if a = 0
N. B. Vyas Laplace Transforms
186. Unit Step function or Heaviside's unit function
Laplace Transform of Unit Step Function:
By de
187. nition of Z Laplace transform
1
Lfu(t a)g =
0
estu(t a)dt
=
Z a
0
est(0)dt +
Z 1
a
est(1)dt
=
Z 1
a
estdt =
est
s
1
a
=
1
s
eas
) L1
1
s
eas
= u(t a)
In particular, if a = 0
L(u(t)) =
1
s
) L1
1
s
= u(t)
N. B. Vyas Laplace Transforms
188. Second Shifting Theorem
Second Shifting Theorem:
If Lff(t)g = f(s), then Lff(t a)u(t a)g = eas f(s)
N. B. Vyas Laplace Transforms
189. Second Shifting Theorem
Second Shifting Theorem:
If Lff(t)g = f(s), then Lff(t a)u(t a)g = eas f(s)
) L1[eas f(s)] = f(t a)u(t a)
Corollary: Lff(t)H(t a)g = easLff(t + a)g
N. B. Vyas Laplace Transforms
190. Second Shifting Theorem
Second Shifting Theorem:
If Lff(t)g = f(s), then Lff(t a)u(t a)g = eas f(s)
) L1[eas f(s)] = f(t a)u(t a)
Corollary: Lff(t)H(t a)g = easLff(t + a)g
Corollary: LfH(t a) H(t b)g =
eas ebs
s
N. B. Vyas Laplace Transforms
191. Second Shifting Theorem
Second Shifting Theorem:
If Lff(t)g = f(s), then Lff(t a)u(t a)g = eas f(s)
) L1[eas f(s)] = f(t a)u(t a)
Corollary: Lff(t)H(t a)g = easLff(t + a)g
Corollary: LfH(t a) H(t b)g =
eas ebs
s
Corollary:
Lff(t) [H(t a) H(t b)]g = easLff(t+a)gebsLff(t+b)g
N. B. Vyas Laplace Transforms