Calculus, 10th Edition (Anton)

Published by Wiley
ISBN 10: 0-47064-772-8
ISBN 13: 978-0-47064-772-1

Chapter 6 - Exponential, Logarithmic, And Inverse Trigonometric Functions - 6.2 Derivatives And Integrals Involving Logarithmic Functions - Exercises Set 6.2 - Page 426: 51

Answer

$$y = \ln \left( {x + 1} \right)$$

Work Step by Step

$$\eqalign{ & {\text{Let the differential equation }}\frac{{dy}}{{dx}} = {e^{ - y}} \cr & {\text{Separate the variables}} \cr & {e^y}dy = dx \cr & {\text{Integrate both sides with respect to }}x \cr & {e^y} = x + C\,\,\left( {\bf{1}} \right) \cr & \cr & {\text{Find the function }}y = f\left( x \right){\text{ with }}y = 0{\text{ when }}x = 0 \cr & {e^0} = 0 + C \cr & C = 1 \cr & \cr & {\text{Substitute }}C = 1{\text{ into equation }}\left( {\bf{1}} \right) \cr & {e^y} = x + 1\,\, \cr & {\text{Solving for }}y \cr & y = \ln \left( {x + 1} \right) \cr} $$
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