Link to actual problem [7548] \[ \boxed {\left (2 x^{2}+1\right ) y^{\prime \prime }-9 x y^{\prime }-6 y=0} \]
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 &= \left (2 x^{2}+1\right )^{\frac {13}{8}} \operatorname {LegendreP}\left (\frac {11}{4}, \frac {13}{4}, i \sqrt {2}\, x \right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {y}{\left (2 x^{2}+1\right )^{\frac {13}{8}} \operatorname {LegendreP}\left (\frac {11}{4}, \frac {13}{4}, i \sqrt {2}\, x \right )}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \left (2 x^{2}+1\right )^{\frac {13}{8}} \operatorname {LegendreQ}\left (\frac {11}{4}, \frac {13}{4}, i \sqrt {2}\, x \right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {y}{\left (2 x^{2}+1\right )^{\frac {13}{8}} \operatorname {LegendreQ}\left (\frac {11}{4}, \frac {13}{4}, i \sqrt {2}\, x \right )}\right ] \\ \end{align*}