\((x^2-yx^2)\dfrac{\mathrm dy}{\mathrm dx}+y^2+xy^2=0\implies \dfrac{y-1}{y^2}\,\mathrm dy=\dfrac{1+x}{x^2}\,\mathrm dx\)
Integrating both sides yields
\(\displaystyle\int\frac{y-1}{y^2}\,\mathrm dy=\int\frac{1+x}{x^2}\,\mathrm dx\)
\(\displaystyle\int\left(\frac1y-\frac1{y^2}\right)\,\mathrm dy=\int\left(\frac1x+\frac1{x^2}\right)\,\mathrm dx\)
\(\ln|y|+\dfrac1y=\ln|x|-\dfrac1x+C\)
