ECE 316 ENGINEERING MATHEMATICS
LECTURE (4) LAPLACE TRANSFORM III
Ayman A. Arafa
ayman.arafa@ejust.edu.eg
Office: B8-F2-04
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OUTLINE
➢ The Inverse Laplace Transform
➢ Laplace Transform and Initial-value Problems
➢ Systems of Linear Differential Equations
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THE INVERSE LAPLACE TRANSFORM
The determination of the inverse Laplace transform of the transform function 𝐹 𝑠 is
not as easier as that of finding the Laplace transform of the function 𝑓 𝑡 .
ℒ
𝑓(𝑡)
𝐹(𝑠)
ℒ −1
𝛼+𝑖∞
1
𝑓 𝑡 =
න 𝑒 𝑠𝑡 𝐹 𝑠 𝑑𝑠
2𝜋𝑖
𝛼−𝑖∞
The evaluation of this integral requires familiarity with contour integration in the
theory of complex functions which is out of the course scope.
THE INVERSE LAPLACE TRANSFORM
Therefore, the inverse Laplace transform will be done using one of the following
inversion methods
1. Inverting using the transform properties.
2. Inverting using partial fractions.
3. The Heaviside expansion theorem.
INVERTING USING THE TRANSFORM PROPERTIES
1
𝑠−𝑎
Γ(𝑛 + 1)
𝑠 𝑛+1
𝑛−1
1
𝑡
ℒ −1 𝑛 =
𝑠
Γ(𝑛)
1
1
−1
ℒ
=
sin 𝑏𝑡
2
𝑠 +𝑏
𝑏
𝑠
−1
ℒ
= cos 𝑏𝑡
𝑠2 + 𝑏
5
INVERTING USING THE TRANSFORM PROPERTIES
a) Linearity: ℒ −1 𝑐1 𝐹 𝑠 + 𝑐2 𝐺 𝑠
= 𝑐1 𝑓 𝑡 + 𝑐2 𝑔(𝑡)
b) First Shifting Property: ℒ −1 𝐹 𝑠 − 𝑎
= 𝑒 𝑎𝑡 𝑓 𝑡
c) Second Shifting Property: ℒ −1 𝑒 −𝑎𝑠 𝐹 𝑠
d) Scaling Property: ℒ −1 𝐹 𝑎𝑠
1
𝑎
= 𝑓
e) Derivative of the transform: ℒ −1 𝐹
f)
Division by 𝑠:
𝐹 𝑠
ℒ −1
𝑠
=
𝑡
‫׬‬0 𝑓
𝑛
𝑡 𝑑𝑡
= 𝑓 𝑡 − 𝑎 𝑈(𝑡 − 𝑎)
𝑡
𝑎
𝑠
= −1 𝑛 𝑡 𝑛 𝑓(𝑡)
INVERTING USING THE TRANSFORM PROPERTIES
Γ(𝑛 + 1)
(𝑠 − 𝑎)𝑛+1
ℒ −1
1
𝑡 𝑛−1 𝑎𝑡
=
𝑒
𝑛
(𝑠 − 𝑎)
Γ(𝑛)
1
1 𝑎𝑡
=
𝑒 sin 𝑏𝑡
2
(𝑠 − 𝑎) +𝑏
𝑏
𝑠−𝑎
−1
𝑎𝑡
ℒ
=
𝑒
cos 𝑏𝑡
(𝑠 − 𝑎)2 +𝑏
1
1 𝑎𝑡
−1
ℒ
=
𝑒 sinℎ 𝑏𝑡
2
(𝑠 − 𝑎) −𝑏
𝑏
ℒ −1
ℒ −1
𝑠−𝑎
𝑎𝑡 cosh 𝑏𝑡
=
𝑒
(𝑠 − 𝑎)2 −𝑏
INVERTING USING THE TRANSFORM PROPERTIES
g) Multiplication by 𝑠: ℒ −1 𝑠𝐹 𝑠
= 𝑓 ′ 𝑡 + 𝑓(0)
∞
h) Integral of the transform: ℒ −1 ‫= 𝑠𝑑 𝑠 𝐹 𝑠׬‬
i)
Convolution Theorem: ℒ −1 𝐹 𝑠 𝐺 𝑠
𝑓 𝑡
𝑡
=𝑓 𝑡 ⋆𝑔 𝑡
EXAMPLES
Find the inverse Laplace Transform of the following
1
1. 𝐹 𝑠 = 𝑠2+6𝑠+9
2𝑠+5
2. 𝐹 𝑠 = (𝑠−3)2
3. 𝐹 𝑠 =
4. 𝐹 𝑠 =
5. 𝐹 𝑠 =
Ans: 𝑓 𝑡 = 𝑡𝑒 −3 𝑡
Ans: 𝑓 𝑡 = 2 + 11 𝑡 𝑒 3 𝑡
1
1 −2 𝑡
Ans: 𝑓 𝑡 = 𝑒
sin 2𝑡
𝑠 2 +4𝑠+8
2
𝑠
1 3𝑡
Ans: 𝑓 𝑡 = 𝑒 2 cos 2𝑡 + 3 2 sin 2𝑡
𝑠 2 −6𝑠+11
2
𝑠 2 +6𝑠+9
1
Ans: 𝑓 𝑡 =
𝑒 −4𝑡 − 96 𝑒 𝑡 + 125𝑒 2𝑡
(𝑠−1)(𝑠−2)(𝑠+4)
30
6. 𝐹 𝑠 =
6𝑠 2 +50
(𝑠+3)(𝑠 2 +4)
7. 𝐹 𝑠 =
𝑡𝑎𝑛−1 𝑠
Ans: 𝑓 𝑡 = 8𝑒 −3 𝑡 − 2 cos 2𝑡 + 3 sin 2𝑡
Ans: 𝑓 𝑡 =
− sin 𝑡
𝑡
EXAMPLES
8. 𝐹 𝑠 =
9. 𝐹 𝑠 =
10. 𝐹 𝑠 =
𝑒 −2𝑠
Ans: 𝑓 𝑡 = 𝑒 4𝑡−8 𝑈(𝑡 − 2)
𝑠−4
𝑠
𝜋
−𝜋𝑠/2
𝑒
Ans:
𝑓
𝑡
=
−
sin
3𝑡
𝑈(𝑡
−
)
𝑠 2 +9
2
𝑠
1
(convolution)
Ans:
𝑓
𝑡
=
𝑡 sin 2𝑡
(𝑠 2 +4)2
4
PROBLEMS
Find the inverse Laplace Transform of the following
1. 𝐹 𝑠 =
2. 𝐹 𝑠 =
𝑠𝑒 −3𝑠
𝑠 2 +16
𝑠−1
𝑠 2 −2𝑠+5
3. 𝐹 𝑠 = ln 1 +
Ans: 𝑓 𝑡 = cos 4 𝑡 − 3 𝑈 𝑡 − 3
Ans: 𝑓 𝑡 = 𝑒 𝑡 cos 2𝑡
1
𝑠
Ans: 𝑓 𝑡 =
1−𝑒 −𝑡
𝑡
4. 𝐹 𝑠 =
1
𝑠 2 +1 2
(convolution)
Ans: 𝑓 𝑡 =
5. 𝐹 𝑠 =
4𝑠−2
𝑠 2 +1 2
(convolution)
Ans: 𝑓 𝑡 = 2 𝑡 sin 𝑡 − sin 𝑡 + 𝑡 cos 𝑡
6. 𝐹 𝑠 =
𝑒 −𝑠 −𝑒 −2𝑠
𝑠2
1
2
sin 𝑡 − 𝑡 cos 𝑡
2
Ans: 𝑓 𝑡 = 𝑡 − 2 𝑈 𝑡 − 2 − 2 𝑡 − 3 𝑈 𝑡 − 3 + 𝑡 − 4 𝑈(𝑡 − 4)
Find the inverse Laplace Transform of the following
3𝑠−2
Ans: 𝑓 𝑡 = 𝑒 2𝑡 (3 cos 4𝑡 + sin 4𝑡)
𝑠+3
Ans: 𝑓 𝑡 = 𝑒 −4𝑡 (1 − 𝑡)
7. 𝐹 𝑠 = 𝑠2−4𝑠+20
8. 𝐹 𝑠 = 𝑠2+8𝑠+16
9. 𝐹 𝑠
10. 𝐹 𝑠
𝑒 −5𝑠
=
𝑠−2 4
𝑠
= 2 2
𝑠 +4
11. 𝐹 𝑠 = ln 1 +
1
𝑠+1 2
4𝑠
𝑠 2 +4 2
2𝑠 2
𝑠 2 +9 2
Ans: 𝑓 𝑡 =
Ans: 𝑓 𝑡 =
1
𝑠2
Ans: 𝑓 𝑡 =
12. 𝐹(𝑠) = 𝑠3
13. 𝐹(𝑠) =
14. 𝐹 𝑠 =
Ans: 𝑓 𝑡 =
(convolution)
(convolution)
1
𝑡 − 5 3 𝑒 2 𝑡−5 𝑈(𝑡 −
6
𝑡 sin 2𝑡
4
2(1−cos 𝑡)
𝑡
1
6 − 4 𝑡 + 𝑡 2 − 2 𝑒 −𝑡
2
Ans: 𝑓 𝑡 = 𝑡 sin 2𝑡
Ans: 𝑓 𝑡 =
1
3
3 𝑡 cos 3𝑡 + sin 3𝑡
5)
3+𝑡
INVERTING USING PARTIAL FRACTIONS.
The transformed function 𝐹 𝑠 is often represented by quotients of polynomials.
Therefore, it is helpful to write a quotient of polynomials as a sum of simpler
quotients known as the partial fractions.
Problems: Find the inverse Laplace transform of the following functions
15. 𝐹 𝑠 =
16. 𝐹 𝑠 =
17. 𝐹 𝑠 =
18. 𝐹 𝑠 =
19. 𝐹 𝑠 =
2𝑠 2 −12
𝑠 3 +4𝑠 2 +3𝑠
48𝑠 2 +96
𝑠 4 −6𝑠 3 +32𝑠
10𝑠 3 −12𝑠−6
𝑠 3 𝑠+1 2
𝑠+10
𝑠 3 −3𝑠 2 +4𝑠−12
3𝑠
𝑠+1 𝑠 2 −2𝑠+5
Ans: 𝑓 𝑡 = −4 + 5𝑒 −𝑡 + 𝑒 −3𝑡
Ans: 𝑓 𝑡 = 3 − 4𝑒 −2𝑡 + 𝑒 4𝑡 + 36𝑡𝑒 4𝑡
Ans: 𝑓 𝑡 = 6 − 3𝑡 2 − 6𝑒 −𝑡 + 4𝑡𝑒 𝑡
Ans: 𝑓 𝑡 = 𝑒 3𝑡 − cos 2𝑡 − sin 2𝑡
Ans: 𝑓 𝑡 =
3
8
𝑒 𝑡 cos 2𝑡 + 3𝑒 𝑡 sin 2𝑡 − 𝑒 −𝑡
STUDY MATERIALS
Book: “Advanced Engineering Mathematics,” Dennis G. Zill & Warren S.
Wright; 5th edition, Jones & Bartlett Learning, 2012
Sections covered
Sections 4.2 and 4.6
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