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: 28


$$y = \frac{1}{n^{2}+1}(n \cos x+ \sin x)+Ce^{-nx}.$$

Work Step by Step

Given $$y^{\prime}+n y=\cos x$$ This is a linear equation with $p(x)=n$ and $q(x) =\cos x$ Since \begin{align*} \mu(x)&=e^{\int p(x)dx}\\ &=e^{\int ndx}\\ &=e^{nx} \end{align*} Then \begin{align*} \mu(x)y&=\int \mu(x)q(x)dx\\ e^{nx} y&=\int e^{nx} \cos xdx \end{align*} Use $$ \int e^{a u} \cos b u d u=\frac{e^{a u}}{a^{2}+b^{2}}(a \cos b u+b \sin b u)+C$$ Then \begin{align*} e^{nx} y&= \frac{e^{nx}}{n^{2}+1}(n \cos x+ \sin x)+C\\ y&= \frac{1}{n^{2}+1}(n \cos x+ \sin x)+Ce^{-nx}. \end{align*}
Update this answer!

You can help us out by revising, improving and updating this answer.

Update this answer

After you claim an answer you’ll have 24 hours to send in a draft. An editor will review the submission and either publish your submission or provide feedback.