Differential Equations and Linear Algebra (4th Edition)

Published by Pearson
ISBN 10: 0-32196-467-5
ISBN 13: 978-0-32196-467-0

Chapter 10 - The Laplace Transform and Some Elementary Applications - 10.1 Definition of the Laplace Transform - Problems - Page 675: 5

Answer

$\dfrac{b}{(s^2-b^2) }$

Work Step by Step

The Laplace Transform can be written as: $L[F(t)]=\int_{0}^{\infty} e^{-st} f(t) dt $ We are given that $f(t)=\sin (h bt)$ Now, $L[F(t)]=\int_{0}^{\infty} e^{-st} f(t) dt \\= \int_{0}^{\infty} e^{-st} [\sin (h bt)] \ dt \\=\int_{0}^{\infty} \dfrac{e^{2st}e^{bt} \ dt}{2}-\int_{0}^{\infty} \dfrac{e^{-st}e^{-bt} \ dt}{2} \\=\dfrac{1}{2(s-b) }-\dfrac{1}{2(s+b)} \\=\dfrac{b}{(s^2-b^2) }$
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