University:
Whitman College
Course:
MATH-244 | Differential Equations
Academic year:

2021
Views:

395

Pages:

1
Author:

Giana Hernandez

1. Use eiat to help compute the Laplace transform of cos(at). SOLUTION: Since eiat = cos(at) + i sin(at), then L(eiat ) = L(cos(at)) + iL(sin(at)) so we compute the Laplace transform of eiat , then take the real part for our answer: iat L(e ) = Z ∞ e −st iat e dt = 0 Z ∞ e−(s−ai)t dt = − 0 0+ 1 e−(s−ai)t s − ai ∞ = 0 1 s + ai = 2 s − ai s + a2 And the real part is s2 /(s2 + a2 ). 2. Same idea, but we’ll compute L(eat (cos(bt) + i sin(bt)). Z ∞ e −st (a+ib)t e dt = Z ∞ e −((s−a)+bi)t 0 0 1 dt = − e−((s−a)−bi)t (s − a) − bi ∞ = 0 (s − a)2 + b2 1 = (s − a) − bi (s − a)2 + b2 We take the real part. 3. Exponential order practice: (a) sin(t): We can use et , for t > 0 (see graph) (b) tan(t) has a vertical asymptote at t = π/2 (and multiples of π thereafter), so it is NOT of exponential order. 3 (c) t3 = eln(t ) = e3 ln(t) ≤ e3t , so this is of exponential order. 2 (d) et is not of exponential order, since the exponent is of order 2. t (e) 5t = eln(5 ) = et ln(5) (f) tt is not of exponential order: t tt = eln(t ) = et ln(t) 1

ComplexLaplaceSOLN

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