Calculus (3rd Edition)

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

Chapter 10 - Introduction to Differential Equations - 10.4 First-Order Linear Equations - Exercises - Page 524: 25


$$ y(x)=\tanh x+3 \operatorname{sech} x$$

Work Step by Step

Given$$y^{\prime}+(\tanh x) y=1, \quad y(0)=3$$ This is a linear equation with $p(x) =\tanh x\ \ q(x) =1$, so \begin{align*} \mu(t)&=e^{\int p(x)dx}\\ &=e^{\int \tanh x d x}\\ &=e^{\ln \cosh x}\\ &=\cosh x \end{align*} Then \begin{align*} y\mu(x) &=\int \mu(x)q(x)dx\\ \cosh x y &=\int \cosh x dx\\ &= \sinh x+C \end{align*} Then $$y(x)=\tanh x+C \operatorname{sech} x$$ Since $y(0)=3 $, then $C=3$, and hence $$ y(x)=\tanh x+3 \operatorname{sech} x$$
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