2.13.2.7 Example 7

\begin{align} y^{\prime }x-y-2\sqrt {xy} & =0\nonumber \\ y^{\prime } & =\frac {y+2\sqrt {xy}}{x} \tag {1}\end{align}

We start by checking if it homogenous or not. Using

\begin{align*} m & =\frac {f+xf_{x}}{f-yf_{y}}\\ & =\frac {\frac {y+2\sqrt {xy}}{x}+x\left ( \frac {y}{x\sqrt {xy}}-\frac {y+2\sqrt {xy}}{x^{2}}\right ) }{\frac {y+2\sqrt {xy}}{x}-y\left ( \frac {1+\frac {x}{\sqrt {xy}}}{x}\right ) }\\ & =1 \end{align*}

Since \(m=1\) then this is homogeneous ode (special case of isobaric). Hence the substitution \(v=\frac {y}{x}\) makes the ode (1) separable.