Example 8 \begin {equation} y^{\prime }=\frac {-y\left ( y^{2}+3x^{2}+2x\right ) }{x^{2}+y^{2}} \tag {1} \end {equation} We start by checking if it homogenous or not. Using\begin {align*} m & =\frac {f+xf_{x}}{f-yf_{y}}\\ & =\frac {\frac {-y\left ( y^{2}+3x^{2}+2x\right ) }{x^{2}+y^{2}}+x\frac {d}{dx}\left ( \frac {-y\left ( y^{2}+3x^{2}+2x\right ) }{x^{2}+y^{2}}\right ) }{\frac {-y\left ( y^{2}+3x^{2}+2x\right ) }{x^{2}+y^{2}}-y\frac {d}{dy}\left ( \frac {-y\left ( y^{2}+3x^{2}+2x\right ) }{x^{2}+y^{2}}\right ) }\\ & =\frac {\frac {-y\left ( y^{2}+3x^{2}+2x\right ) }{x^{2}+y^{2}}+x\left ( -2\frac {y\left ( -x^{2}+2xy^{2}+y^{2}\right ) }{\left ( x^{2}+y^{2}\right ) ^{2}}\right ) }{\frac {-y\left ( y^{2}+3x^{2}+2x\right ) }{x^{2}+y^{2}}-y\left ( -\frac {3x^{4}+2x^{3}-2xy^{2}+y^{4}}{\left ( x^{2}+y^{2}\right ) ^{2}}\right ) }\\ & =\frac {3x^{4}+8x^{2}y^{2}+4xy^{2}+y^{4}}{4x^{2}y^{2}+4xy^{2}} \end {align*} Since this does not simplify to numerical value, it is not homogenous ode. This turns out to be homogenous type D. See earlier note on this. There is a slight difference in definition between homogenous ode and homogenous type D. In Maple terms, homogenous ode is called homogenous ode type A. A homogenous type D is one in which the substitution \(y=ux\) makes the ode separable or quadrature.