Calculus (3rd Edition)

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

Chapter 10 - Introduction to Differential Equations - Chapter Review Exercises - Page 526: 29


$y=- \cos x +\frac{1}{x}\sin x +\frac{C}{x}$

Work Step by Step

This is a linear equation and has the integrating factor as follows $$\alpha(x)= e^{\int P(x)dx}=e^{ \int \frac{1}{x} dx}=e^{\ln x}=x.$$ Now the general solution is \begin{align} y& =\alpha^{-1}(x)\left( \int\alpha(x) Q(x)dx +C\right)\\ & =\frac{1}{x}\left( -x\cos x +\sin x +C\right)\\ & =- \cos x +\frac{1}{x}\sin x +\frac{C}{x}\\ \end{align} where we did the integration by parts.
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