Link to actual problem [8081] \[ \boxed {2 x^{2} \left (3 x +2\right ) y^{\prime \prime }+x \left (4+21 x \right ) y^{\prime }-\left (1-9 x \right ) y=0} \]
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 {\sqrt {x}}{\left (3 x +2\right )^{\frac {3}{2}}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {\left (3 x +2\right )^{\frac {3}{2}} y}{\sqrt {x}}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \frac {\sqrt {2}}{\left (3 x +2\right ) \sqrt {x}}+\frac {3 \sqrt {x}\, \operatorname {arctanh}\left (\frac {\sqrt {3 x +2}\, \sqrt {2}}{2}\right )}{\left (3 x +2\right )^{\frac {3}{2}}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {y}{\frac {\left (9 x^{2}+6 x \right ) \operatorname {arctanh}\left (\frac {\sqrt {3 x +2}\, \sqrt {2}}{2}\right )}{\left (3 x +2\right )^{\frac {5}{2}} \sqrt {x}}+\frac {\sqrt {2}}{\left (3 x +2\right ) \sqrt {x}}}\right ] \\ \end{align*}