Calculus (3rd Edition)

Published by W. H. Freeman
ISBN 10: 1464125260
ISBN 13: 978-1-46412-526-3

Chapter 10 - Introduction to Differential Equations - 10.1 Solving Differential Equations - Exercises - Page 505: 16


The general solution is $$ y=\ln \frac{1}{c-e^x}.$$

Work Step by Step

We have $$ y'-e^{x+y}=0\Longrightarrow e^{-y}dy=e^{x}dx\\ \Longrightarrow\int e^{-y}dy=\int e^{x}dx+c\\ \Longrightarrow -e^{-y} =e^{x}+c\\ \Longrightarrow e^{-y} =c-e^{x}\Longrightarrow -y =\ln(c-e^x).$$ Hence, the general solution is $$ y=\ln \frac{1}{c-e^x}.$$
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