Calculus, 10th Edition (Anton)

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

Chapter 8 - Mathematical Modeling With Differential Equations - 8.4 First-Order Differential Equations And Applications - Exercises Set 8.4 - Page 592: 6


$$y=e^{-x } \ln |e^x-1|+ce^{-x } $$

Work Step by Step

Given $$ \frac{dy}{dx} + y+\frac{1}{1-e^x}=0$$ Rewriting equation $$ \frac{dy}{dx} + y=\frac{1}{e^x-1}$$ Since \begin{align*} \mu(x)&=e^{\int dx}\\ &=e^{x } \end{align*} Then \begin{align*} y\mu(x)&=\int \mu(x) q(x)dx\\ e^{x } y&=\int \frac{e^{x }}{e^x-1} dx\\ &=\ln |e^x-1|+c \end{align*} Hence $$y=e^{-x } \ln |e^x-1|+ce^{-x } $$
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