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Quiz

# quiz 4 (10).pdf

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Queen's University

Mathematics & Engineering Courses

MTHE 225

Gregory G Smith

Fall

Description

Quiz 4a Solutions
1. (a) Using only the deﬁnition of the Laplace transform, ﬁnd the Laplace transform
of f(t) where
2t
e 0 ≤ t ≤ 3
f(t) =
0 t > 3
We compute:
▯ ∞
L {f(t)} = f(t)estdt
▯ 03 ▯ ∞
−st −st
= f(t)e dt + f(t)e dt
0 3
▯ 3 ▯ ∞
= e e −sdt + 0e−stdt
0 3
▯ 3
(2−s)t
= e dt
0 ▯
1 (2−s)t
= e ▯
2 − s 0
1 3(2−s) 1 0
= e − e
2 − s 2 − s
e3(2−s− 1
=
2 − s
2. It should be clear that the expression for F(s) you found in part (a) is undeﬁned
at a particular value of s. But we may use the deﬁnition of the Laplace transform
to compute F(s) at this particular value. Do so.
F(s) is undeﬁned at s = 2, due to the vanishing denominator. However, from the
deﬁnition of the Laplace transform,
▯ ∞
−2t
F(2) = f(t)e dt
▯0 ▯
3 2t −2t ∞ −st
= e e dt + 0e dt
0 3
▯ 3
= dt
0

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