\[ xyy^{\prime \prime }+x\left ( y^{\prime }\right ) ^{2}-yy^{\prime }=0 \] Integrating both sides gives\begin {align*} \int xyy^{\prime \prime }+x\left ( y^{\prime }\right ) ^{2}-yy^{\prime }dx & =c_{1}\\ xyy^{\prime }-y^{2} & =c_{1}\\ y^{\prime } & =\frac {c_{1}}{xy}+\frac {y}{x}\\ & =\frac {c_{1}+y^{2}}{xy}\\ & =\left ( \frac {c_{1}+y^{2}}{y}\right ) \frac {1}{x} \end {align*}
Which is separable and easily solved.