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MTHE 225 (6)
Quiz

quiz 4 (10).pdf

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School
Queen's University
Department
Mathematics & Engineering Courses
Course
MTHE 225
Professor
Gregory G Smith
Semester
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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