Calculus 8th Edition

Published by Cengage
ISBN 10: 1285740629
ISBN 13: 978-1-28574-062-1

Chapter 6 - Inverse Functions - 6.2 Exponential Functions and Their Derivatives - 6.2 Exercises: 56

Answer

$y''+2y'+y=0$

Work Step by Step

Show that the function $y=Ae^{-x}+Bxe^{-x}$ satisfies the differential equation $y''+2y'+y=0$. $y'=-Ae^{-x}-Bxe^{-x}+Be^{-x}$ and $y''=Ae^{-x}-Bxe^{-x}-2Be^{-x}$ Now, $y''+2y'+y$ $=Ae^{-x}-Bxe^{-x}-2Be^{-x}+2(-Ae^{-x}-Bxe^{-x}+Be^{-x})+Ae^{-x}+Bxe^{-x}$ $=0$ Hence, the function $y=Ae^{-x}+Bxe^{-x}$ satisfies the differential equation $y''+2y'+y=0$.
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