Integrate the differential equation:
![\[ y' = 1 + \frac{y}{x}. \]](https://www.stumblingrobot.com/wp-content/ql-cache/quicklatex.com-212b81a9417a3f4f67b69e5a1797e4b4_l3.png "Rendered by QuickLaTeX.com")
We use the substitution 
(used in the examples of Section 8.25 of Apostol). So we have
![\[ y = vx \quad \implies \quad y' = v + v'x. \]](https://www.stumblingrobot.com/wp-content/ql-cache/quicklatex.com-54a2f011fd547873a5ed8bc4f75a72be_l3.png "Rendered by QuickLaTeX.com")
(Where the derivative 
is taken with respect to 
.) Making this substitution in the given equation (and noting that implies 
) we have
![\begin{align*} y' = 1 + \frac{y}{x} && \implies && v + v'x &= 1 + v \\ && \implies && v' &= \frac{1}{x} \\[9pt] && \implies && \int dv &= \int \frac{1}{x} \, dx \\[9pt] && \implies && v &= \log |x| + C \\ && \implies && y &= x \log |x| + Cx. \end{align*}](https://www.stumblingrobot.com/wp-content/ql-cache/quicklatex.com-52c7db003d753a6224e77942ef208a71_l3.png "Rendered by QuickLaTeX.com")