Calculus 8th Edition

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

Chapter 6 - Inverse Functions - 6.3* The Natural Exponential Function - 6.3* Exercises: 57

Answer

$2y''-y'-y=0$

Work Step by Step

$y=e^x+e^{-x/2}$ By the chain rule: $y'=e^x+e^{-x/2}*\frac{d}{dx}(-\frac{x}{2})$ $y'=e^x-\frac{1}{2}e^{-x/2}$ $y''=e^x-\frac{1}{2}e^{-x/2}*\frac{d}{dx}(-\frac{x}{2})$ $y''=e^x+\frac{1}{4}e^{-x/2}$ $2y''-y'-y$ $=2(e^x+\frac{1}{4}e^{-x/2})-(e^x-\frac{1}{2}e^{-x/2})-(e^x+e^{-x/2})$ $=2e^x+\frac{1}{2}e^{-x/2}-e^x+\frac{1}{2}e^{-x/2}-e^x-e^{-x/2}$ $=0$
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