Using only the Laplace transform table given to obtain the Laplace transform of the following functions: (a) cosh(t) + cos(t). (b) 3e- + 2 - 2 sin(2t). -5t 2 The function "cosh" stands for hyperbolic cosine and cosh(x) e+e. The results must be written as a single rational function and be simplified whenever possible. Showing result only without reasoning or argumentation will be insufficient. =
Using only the Laplace transform table given to obtain the Laplace transform of the following functions: (a) cosh(t) + cos(t). (b) 3e- + 2 - 2 sin(2t). -5t 2 The function "cosh" stands for hyperbolic cosine and cosh(x) e+e. The results must be written as a single rational function and be simplified whenever possible. Showing result only without reasoning or argumentation will be insufficient. =
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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Transcribed Image Text:Using only the Laplace transform table given to obtain the Laplace transform of the following functions:
(a) cosh(t) + cos(t).
(b) 3e-5t+2- 2 sin(2t).
The function "cosh" stands for hyperbolic cosine and cosh(x)
The results must be
written as a single rational function and be simplified whenever possible. Showing result only without
reasoning or argumentation will be insufficient.
=
e te
2
Transcribed Image Text:(a) Table of Laplace
transform pairs;
(b) some properties of
the Laplace transform.
(a)
(b)
The Laplace transform table
f(t)
c, c a constant
t
t", n a positive integer
ekt, k a constant
sin at, a a real constant
cos at, a a real constant
e-kr
kt sin at, k and a real constants
e-kt cos at, k and a real constants
Linearity:
First shift theorem:
L{f(t)} = F(s)
Derivative of transform:
C
12-13
n!
S
2+1
2
-k
a
2
+ a
S
s² + a²
2
(s+k)² + a²
s+k
2
(s+k)² +a
Region of convergence
Re(s) > 0
Re(s) > 0
Re(s) > 0
Re(s) > Re(k)
Re(s) > 0
Re(s) > 0
Re(s) > -k
L{f(t)} = F(s), Re(s) >₁ and L{g(t)} = G(s), Re(s) > 0₂
L{ af(t) + Bg(t)} = aF(s) + BG(s), Re(s) > max(σ₁,0₂)
Leat f(t)} = F(sa), Re(s) > σ₁ + Re(a)
Re(s) > -k
L{t" f(t)} = (-1)d"F(s), (n=1,2,...), Re(s) > 0₁
ds"
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