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: 5


$$y=x^{-1}\left( \frac{x^2}{2} +C\right).$$

Work Step by Step

Rewriting the equation as $$y'+x^{-1}y=1$$ This is a linear equation and has the integrating factor as follows $$\alpha(x)= e^{\int P(x)dx}=e^{\int x^{-1} dx}=e^{\ln x}=x$$ Now the general solution is \begin{align} y&=\alpha^{-1}(x)\left( \int \alpha(x)Q(x)dx +C\right)\\ &=x^{-1}\left( \int x dx +C\right)\\ &=x^{-1}\left( \frac{x^2}{2} +C\right) \end{align} The general soluiton is $$y=x^{-1}\left( \frac{x^2}{2} +C\right).$$
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