Integrate the differential equation:
![\[ y' = \frac{x^2 + 2y^2}{xy}. \]](https://www.stumblingrobot.com/wp-content/ql-cache/quicklatex.com-9af79af7fb62f939526a0deaec308a82_l3.png "Rendered by QuickLaTeX.com")
Like in the previous exercise (Section 8.26, Exercise #3) we make the substitution 
which implies 
. Therefore,
![\begin{align*} y' = \frac{x^2 + 2y^2}{xy} && \implies && y' &= \frac{x}{y} + 2 \frac{y}{x} \\ && \implies && v + v'x &= \frac{1}{v} + 2v \\[9pt] && \implies && v' \left( \frac{v}{1+v^2} \right) &= \frac{1}{x} \\[9pt] && \implies && \int \frac{v}{1+v^2} \, dv &= \int \frac{1}{x} \, dx \\[9pt] && \implies && \frac{1}{2} \log (1+v^2) &= \log |x| + C \\[9pt] && \implies && \log \left( \frac{x^2+y^2}{x^2} \right) &= \log (x^2) + C \\[9pt] && \implies && \log (x^2+y^2) &= \log (x^4) + C \\[9pt] && \implies && x^2 + y^2 &= Cx^4. \end{align*}](https://www.stumblingrobot.com/wp-content/ql-cache/quicklatex.com-860747c4318cba1d77f7ed2d6ce07104_l3.png "Rendered by QuickLaTeX.com")
Integrate the differential equation:
Like in the previous exercise (Section 8.26, Exercise #3) we make the substitution which implies . Therefore,