4.4.4.1 Example 1

\[ xyy^{\prime \prime }+x\left ( y^{\prime }\right ) ^{2}-yy^{\prime }=0 \]

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.