Example 6 \begin {align} \left ( x-y\right ) y^{\prime }-x-y & =0\nonumber \\ y^{\prime } & =\frac {x+y}{x-y}\tag {1}\\ & =f\left ( x,y\right ) \nonumber \end {align} We start by checking if it homogenous or not. Using\begin {align*} m & =\frac {f+xf_{x}}{f-yf_{y}}\\ & =\frac {\frac {x+y}{x-y}+x\left ( \frac {1}{x-y}-\frac {x+y}{\left ( x-y\right ) ^{2}}\right ) }{\frac {x+y}{x-y}-y\left ( \left ( \frac {1}{x-y}+\frac {x+y}{\left ( x-y\right ) ^{2}}\right ) \right ) }\\ & =\frac {x\left ( \frac {1}{x-y}-\frac {x+y}{\left ( x-y\right ) ^{2}}\right ) }{-y\left ( \left ( \frac {1}{x-y}+\frac {x+y}{\left ( x-y\right ) ^{2}}\right ) \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.