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: 18

Answer

$$\dfrac{6}{s^3}-\dfrac{5s}{s^2-4}+\dfrac{3}{s^2-9}$$

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 $f(t)=3 t^2 -5 \cos 2t +\sin (3t)$ Now, $L[F(t)]=L[3 t^2 -5 \cos 2t +\sin (3t)] \\=L[3t^2]-L[5 \cos 2t]+L[\sin (3t)] \\=\dfrac{(3)(2)}{s^3}-\dfrac{(5)(s)}{s^2+4}+\dfrac{3}{s^2+9}\\=\dfrac{6}{s^3}-\dfrac{5s}{s^2-4}+\dfrac{3}{s^2-9}$
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