Link to actual problem [8079] \[ \boxed {x^{2} \left (-2 x^{2}+1\right ) y^{\prime \prime }+x \left (-13 x^{2}+7\right ) y^{\prime }-14 y x^{2}=0} \]
type detected by program
{"kovacic"}
type detected by Maple
[[_2nd_order, _with_linear_symmetries]]
Maple symgen result This shows Maple’s found \(\xi ,\eta \) and the corresponding canonical coordinates \(R,S\)\begin{align*} \\ \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \frac {\left (2 x^{2}-1\right )^{\frac {5}{4}}}{x^{6}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {x^{6} y}{\left (2 x^{2}-1\right )^{\frac {5}{4}}}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \frac {5 x^{4}-20 x^{2}+8}{x^{6}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {x^{6} y}{5 x^{4}-20 x^{2}+8}\right ] \\ \end{align*}