5 3. Use partial fraction decomposition, linearity and s-shifting theorem to find the inverse Laplace transform of the function H(s) s(s² + 2s + 2)'
5 3. Use partial fraction decomposition, linearity and s-shifting theorem to find the inverse Laplace transform of the function H(s) s(s² + 2s + 2)'
Trigonometry (MindTap Course List)
10th Edition
ISBN:9781337278461
Author:Ron Larson
Publisher:Ron Larson
Chapter3: Additional Topics In Trigonometry
Section: Chapter Questions
Problem 41CT
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Transcribed Image Text:The charge on a capacitor q(t) in a circuit with a resistor, a capacitor and an inductor connected
in series driven by a given time-dependent voltage v(t) is governed by the second-order ODE
d"(t) + 2q'(t) + 2q(t) = v(t).
(1)
A schematic of the RLC circuit can be found in the course notes.
In Questions 1 - 6 you will calculate Laplace transforms / inverse Laplace transforms of func-
tions. These calculations will be useful in determining the response of (1) for an initial value
problem specified in Question 7.
You MUST use the method of solution as per the instructions found in each of the below
questions.
1 1. Use the linearity of property of Laplace transforms and the table of Laplace transforms on
MyUni to find the Laplace transform of
v(t) = av1 (t) + Bv2(t),
where vi(t) = 6(t – 2) and v2(t) = u(t – 3). Here, a and ß are constants, and 6(t) and u(t)
are the Dirac delta function and unit step function.
2 2. Complete the square of the denominator and use the s-shifting theorem to find the inverse
Laplace transform of the function
F(s) =
82 + 28 + 2°
Where applicable, you may use entries (1) through to (15) and the basic general formulae
from the table of Laplace transforms on MyUni. You must NOT use any entries (16)
through to (23).
5 3. Use partial fraction decomposition, linearity and s-shifting theorem to find the inverse
Laplace transform of the function
Н() —
s(s² + 2s + 2)*
Where applicable, you may use entries (1) through to (15) and the basic general formulae
from the table of Laplace transforms on MyUni. You must NOT use any entries (16)
through to (23).
Transcribed Image Text:| 25 |**
| OLt - a)
1:4
1:4
Table of Laplace Transforms
F(s) = L{S(t)}
f(t)
1/s
1/s²
1/s" (n = 1,2, ...)
| 1/V
1/s3/2
1/s (a > 0)
1
1
t"-1/(n – 1)!
1//mt
3
4
Basic General Formulas for the Laplace Transform
6.
t0-1/T(a)
1
Formula
Name, Comments
eat
8- a
1
F(s) = L{S(1} = [e"()dt
Definition of Transform
teat
(8 – a)²
1
| (1) = c-'{F(»)}
Inverse Transform
9.
(8 – a)™
1
(n = 1, 2, .)
(n-1)!
1
(eat – e")
C{af(t) + bg(t)} = aL{f(t)} + bL{g(t)}
Linearity
1
(a # b)
10
(8 – a)(s – b)
(a – b)
1
(aet – be")
- sCf) – f(0)
(a ± b)
11
(8 – a
(s – b)
(a – b)
Differentiation of Function
- PL(S) – sf(0) – s'(0)
12
sin wt
s² + w?
cism) = "L(J) – *-'s(0) – .. – fln-1)(0)
13
Cos wt
C{e“{(t)} = F(s – a)
-Shifting
a
14
sinh at
L-{F(s – a)} = e" S()
(Ist Shifting Theorem)
15
cosh at
L{{(t – a)u(t – a)} = e~**F(s)
t-Shifting
at sin wt
16
(8 – a)² + w²
{e"F(s)} = f(t – a)u(t – a)
(2nd Shifting Theorem)
8-a
17
eat cos wt
(S • 9)(t) =
[(r)a(t – r)dr
(6 – a)² + w²
Convolution
1.
(1 – cos wt)
18
s(s² + w²)
CJ • 9) - C)C()
19
(wt – sin wt)
1
(sin wt – wt cos wt)
2
1
20
(8² + w?)Z
sin wt
21
(s² + w?)Z
1
(sin wt + wt coswt)
22
(s² + wa )Z
(a² # b³)
1
(cos at - cos bt)
23
(s² + a²)(s² + b²)
62 - a
24
u(t – a)
25
eas
6(t – a)
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