(f) Determine whether or not the following polynomials are stable: (i) s2 - 2s + 4 (ii) + 2s2 + 4 (iii) s3 + 2s2 + s +4 You may freely use any results from lectures, but should state which results you are using and how you are using them.
(f) Determine whether or not the following polynomials are stable: (i) s2 - 2s + 4 (ii) + 2s2 + 4 (iii) s3 + 2s2 + s +4 You may freely use any results from lectures, but should state which results you are using and how you are using them.
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
part F
Transcribed Image Text:(a) State the definition of the Laplace transform of a locally integrable function
f: (0, 00) → C.
(b) Let a € R. Show that the Laplace transform of the function f : [0, 00) → C defined
by f(t) = sin (at) is equal to
3.
2a2
s(s? + 4a?)
for s € Co.
(c) Suppose thatf : (0, 00) C has Laplace transform L{f} = f, with domain Cg
for some B E R. Given y, 8 > 0, find expressions for the Laplace transforms of the
related functions
%3D
(i) t- f(t/y)
(ii) t+ f(t – 8)H(t - 6)
stating carefully a valid domain for each Laplace transform. Here H denotes the
step function H : R R, H(t) = 0 ift < 0 and H(t) = 1 if t20.
(d) Solve the following scalar initial value problem by using the Laplace transform
2 (t) – 32(t) – 6ż(t) + 8z(t) = t, t> 0, (0) = ż(0) = z(0) = 0.
(e) State the definition of a stable polynomial, and give a reason from a
differential equations or control theory perspective why stability of polynomials is
important.
(f) Determine whether or not the following polynomials are stable:
(i) s? - 2s + 4
(ii) 3 + 25? + 4
(iii) s3 + 2s? +s +4
You may freely use any results from lectures, but should state which results you are
using and how you are using them.
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