Find the Inverse Laplace transforms of the following functions: (4s + 2)/(s2 +6s + 34) 1/ [(s-2)(s-4)(s-8)]
Find the Inverse Laplace transforms of the following functions: (4s + 2)/(s2 +6s + 34) 1/ [(s-2)(s-4)(s-8)]
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
Find the Inverse Laplace transforms of the following functions:
- (4s + 2)/(s2 +6s + 34)
- 1/ [(s-2)(s-4)(s-8)]
Subject : DIFFERENTIAL EQUATION
Topic: Inverse Laplace Transforms
Below is the Table of Laplace Transforms.
![1.
3.
5.
7.
9.
f(t)=2²¹{F(s)}
1
19.
t", n=1,2,3,...
11. sin(at)-at cos(at)
21.
√t
sin(at)
t sin (at)
13. cos(at)-at sin (at)
15. sin(at+b)
17. sinh(ar)
et sin (br)
e” sinh(br)
23. te, n=1,2,3,...
u(t)= u(t-c)
Heaviside Function
25.
27. u(t)f(t-c)
29. eºf(t)
31. ƒ(1)
33. ff(t-1)g(1) dr
35. f'(t)
37. f(t)
Table of Laplace Transforms
F(s) = {f(t)}
S
L
su+1
a
s² + a²
2as
(s² + a²)²
2a³
(s² + a²)²³
s(s²_a²)
(s² + a²)²
s sin (b) + acos (b)
s² + a²
3²-d²
b
(s-a)² + b²
b
(s-a)²-B²
n!
(s-a)***
S
e¯ F (s)
F(s-c)
*F(u) du
2.
f(t)=2²¹{F(s)}
at
4. tº,p>-1
e
6., n=1,2,3,...
8. cos(at)
10. tcos(at)
20.
12. sin (at) + at cos(at)
22.
14. cos(at)+ at sin(at)
16. cos(at+b)
18.
cosh (at)
e cos (bt)
ea cosh (br)
24. f(ct)
8(t-c)
Dirac Delta Function
u. (t)g(t)
26.
28.
30. tf(t), n=1,2,3,...
32. ff(v) dv
F(s) G(s)
SF (s)-f(0) 36. f(t)
34. f(t+T)=f(t)
F(s) = £{f(t)}
1
s-a
[(p+1)
Sit
1-3-5---(2n-1)√√
2" 5"
S
s² + a²
s²-a²
(s² + a²)²
2as²
(s² + a²)²
s(s²+3a²)
(s² + a²)²
scos (b)-a sin (b)
s² + a²
S
3-0²
s-a
(s-a)² + b²
s-a
(s-a)²³-5²
e{g(t+c)}
(-1)" F(") (s)
F(s)
S
fe" f (t) dt
1-e
s²F (s)-sf (0)-f(0)
s"F(s)-5-¹ƒ (0)-5-²ƒ' (0)---sf("~²) (0) — ƒ(¹)(0)
-](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F9e6e5c94-76b9-42a9-ad2f-062a3b0480df%2F7f0e26c1-7b9c-4a7b-b676-5f4d32fc791b%2F5dm8b5k_processed.png&w=3840&q=75)
Transcribed Image Text:1.
3.
5.
7.
9.
f(t)=2²¹{F(s)}
1
19.
t", n=1,2,3,...
11. sin(at)-at cos(at)
21.
√t
sin(at)
t sin (at)
13. cos(at)-at sin (at)
15. sin(at+b)
17. sinh(ar)
et sin (br)
e” sinh(br)
23. te, n=1,2,3,...
u(t)= u(t-c)
Heaviside Function
25.
27. u(t)f(t-c)
29. eºf(t)
31. ƒ(1)
33. ff(t-1)g(1) dr
35. f'(t)
37. f(t)
Table of Laplace Transforms
F(s) = {f(t)}
S
L
su+1
a
s² + a²
2as
(s² + a²)²
2a³
(s² + a²)²³
s(s²_a²)
(s² + a²)²
s sin (b) + acos (b)
s² + a²
3²-d²
b
(s-a)² + b²
b
(s-a)²-B²
n!
(s-a)***
S
e¯ F (s)
F(s-c)
*F(u) du
2.
f(t)=2²¹{F(s)}
at
4. tº,p>-1
e
6., n=1,2,3,...
8. cos(at)
10. tcos(at)
20.
12. sin (at) + at cos(at)
22.
14. cos(at)+ at sin(at)
16. cos(at+b)
18.
cosh (at)
e cos (bt)
ea cosh (br)
24. f(ct)
8(t-c)
Dirac Delta Function
u. (t)g(t)
26.
28.
30. tf(t), n=1,2,3,...
32. ff(v) dv
F(s) G(s)
SF (s)-f(0) 36. f(t)
34. f(t+T)=f(t)
F(s) = £{f(t)}
1
s-a
[(p+1)
Sit
1-3-5---(2n-1)√√
2" 5"
S
s² + a²
s²-a²
(s² + a²)²
2as²
(s² + a²)²
s(s²+3a²)
(s² + a²)²
scos (b)-a sin (b)
s² + a²
S
3-0²
s-a
(s-a)² + b²
s-a
(s-a)²³-5²
e{g(t+c)}
(-1)" F(") (s)
F(s)
S
fe" f (t) dt
1-e
s²F (s)-sf (0)-f(0)
s"F(s)-5-¹ƒ (0)-5-²ƒ' (0)---sf("~²) (0) — ƒ(¹)(0)
-
![Table Notes
1. This list is not a complete listing of Laplace transforms and only contains some of
the more commonly used Laplace transforms and formulas.
2. Recall the definition of hyperbolic functions.
e' + e
cosh (
2
sinh
3. Be careful when using "normal" trig function vs. hyperbolic functions. The only
difference in the formulas is the "+ a²" for the "normal" trig functions becomes a
"- a*** for the hyperbolic functions!
4. Formula #4 uses the Gamma function which is defined as
r(t)=x²²dx
If n is a positive integer then,
T(n+1)=n!
The Gamma function is an extension of the normal factorial function. Here are a
couple of quick facts for the Gamma function
I(p+1)=pr (p)
p(p+1)(p+2).(p+n−1)=-
(2)=√
I(p+n)
I(p)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F9e6e5c94-76b9-42a9-ad2f-062a3b0480df%2F7f0e26c1-7b9c-4a7b-b676-5f4d32fc791b%2Fq9kfxal_processed.png&w=3840&q=75)
Transcribed Image Text:Table Notes
1. This list is not a complete listing of Laplace transforms and only contains some of
the more commonly used Laplace transforms and formulas.
2. Recall the definition of hyperbolic functions.
e' + e
cosh (
2
sinh
3. Be careful when using "normal" trig function vs. hyperbolic functions. The only
difference in the formulas is the "+ a²" for the "normal" trig functions becomes a
"- a*** for the hyperbolic functions!
4. Formula #4 uses the Gamma function which is defined as
r(t)=x²²dx
If n is a positive integer then,
T(n+1)=n!
The Gamma function is an extension of the normal factorial function. Here are a
couple of quick facts for the Gamma function
I(p+1)=pr (p)
p(p+1)(p+2).(p+n−1)=-
(2)=√
I(p+n)
I(p)
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