4.13.2.9 Example 9

\begin{equation} y^{\prime }=\frac {\left ( -108y^{2}+12\sqrt {-108y^{3}x^{3}+81y^{4}}\right ) ^{\frac {2}{3}}+12xy}{6\left ( -108y^{2}+12\sqrt {-108y^{3}x^{3}+81y^{4}}\right ) ^{\frac {1}{3}}} \tag {1}\end{equation}

We start by checking if it homogenous or not. Using

\[ m=\frac {f+xf_{x}}{f-yf_{y}}\]

Which simplifies to

\[ m=3 \]

Hence the substitution \(y=vx^{m}\) will make the ode separable. Substituting \(y=vx^{3}\) in (1) results in separable ode. But for this case, we have to assume \(x>0\) in order to simplify it. The resulting ode is too long to write now, but verified to be separable using the computer.