Link to actual problem [9640] \[ \boxed {x \left (x^{2}+1\right ) y^{\prime \prime }+\left (2 \left (1+n \right ) x^{2}+2 n +1\right ) y^{\prime }-\left (v -n \right ) \left (v +n +1\right ) x y=0} \]
type detected by program
{"unknown"}
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 &= x^{-n} \operatorname {LegendreP}\left (v , n , \sqrt {x^{2}+1}\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {x^{n} y}{\operatorname {LegendreP}\left (v , n , \sqrt {x^{2}+1}\right )}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= x^{-n} \operatorname {LegendreQ}\left (v , n , \sqrt {x^{2}+1}\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {x^{n} y}{\operatorname {LegendreQ}\left (v , n , \sqrt {x^{2}+1}\right )}\right ] \\ \end{align*}