11. Use the shift property to find the Laplace transform of eat sinkt.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Question
Please do 11
![146
10. Sketch the graphs of sin t, sin(t-T/2), and H(t-T/2) sin(t-/2), and
find the Laplace transform of each.
11. Use the shift property to find the Laplace transform of eat sinkt.
12. Use the switching property to find the Laplace transform of
x(t) = { [0
13. Show that
e-t,
L [ƒ(at)] = = F (²),
a
∞
t<2
t> 2.
a) Does the function r(t) = et² have a Laplace transform? (b) What
out r(t) = 1/t? Explain why or why not. (c) State why X(s) =
anot be the Laplace transform of some function x = x(t).
Plot the square-wave function
n=0
, a > 0.
f(t) = [(-1)"H(t − n)
3. Laplace Transforms
∞
on the interval t≥ 0 and find its transform F(s). Hint: Use the geometric
1/(1-2) to find the sum.
series 1+ 2+z²+...
16. From the definition of the Laplace transform, find L1/√t using the inte
gral substitution st = r² and using fo exp(-²) dr = √√/2.
17. The Gamma function is a special function defined by
[(y) =
e-tu-¹dt, y> -1.
It is important in probability, statistics, and many other areas of mathe
matics, science, and engineering.
a) Show that I(n+1) = n(n) and I'(n+1) = n! for nonnegative integers](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc10cce41-ef51-4545-a724-9736e236d3b7%2F61fff2b7-74f0-43c6-8e96-ca73fafa0018%2Fr53s79m_processed.jpeg&w=3840&q=75)
Transcribed Image Text:146
10. Sketch the graphs of sin t, sin(t-T/2), and H(t-T/2) sin(t-/2), and
find the Laplace transform of each.
11. Use the shift property to find the Laplace transform of eat sinkt.
12. Use the switching property to find the Laplace transform of
x(t) = { [0
13. Show that
e-t,
L [ƒ(at)] = = F (²),
a
∞
t<2
t> 2.
a) Does the function r(t) = et² have a Laplace transform? (b) What
out r(t) = 1/t? Explain why or why not. (c) State why X(s) =
anot be the Laplace transform of some function x = x(t).
Plot the square-wave function
n=0
, a > 0.
f(t) = [(-1)"H(t − n)
3. Laplace Transforms
∞
on the interval t≥ 0 and find its transform F(s). Hint: Use the geometric
1/(1-2) to find the sum.
series 1+ 2+z²+...
16. From the definition of the Laplace transform, find L1/√t using the inte
gral substitution st = r² and using fo exp(-²) dr = √√/2.
17. The Gamma function is a special function defined by
[(y) =
e-tu-¹dt, y> -1.
It is important in probability, statistics, and many other areas of mathe
matics, science, and engineering.
a) Show that I(n+1) = n(n) and I'(n+1) = n! for nonnegative integers
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