Link to actual problem [9589] \[ \boxed {x \left (x +2\right ) y^{\prime \prime }+2 \left (n +1+\left (n +1-2 l \right ) x -l \,x^{2}\right ) y^{\prime }+\left (2 l \left (p -n -1\right ) x +2 p l +m \right ) y=0} \]
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 &= \left (2+x \right )^{-\frac {1}{2}-\frac {n}{2}} \left (-1-\frac {x}{2}\right )^{\frac {n}{2}+\frac {1}{2}} \operatorname {HeunC}\left (4 l , n , n , -4 p l , 2 l n +2 p l +2 l -\frac {1}{2} n^{2}+m -n , -\frac {x}{2}\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {2 \left (2+x \right )^{\frac {n}{2}} \sqrt {2+x}\, \left (-1-\frac {x}{2}\right )^{-\frac {n}{2}} y}{\sqrt {-4-2 x}\, \operatorname {HeunC}\left (4 l , n , n , -4 p l , 2 \left (n +1+p \right ) l -\frac {n^{2}}{2}+m -n , -\frac {x}{2}\right )}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= x^{-n} \left (2+x \right )^{-\frac {1}{2}-\frac {n}{2}} \left (-1-\frac {x}{2}\right )^{\frac {n}{2}+\frac {1}{2}} \operatorname {HeunC}\left (4 l , -n , n , -4 p l , 2 l n +2 p l +2 l -\frac {1}{2} n^{2}+m -n , -\frac {x}{2}\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {2 x^{n} \left (2+x \right )^{\frac {n}{2}} \sqrt {2+x}\, \left (-1-\frac {x}{2}\right )^{-\frac {n}{2}} y}{\sqrt {-4-2 x}\, \operatorname {HeunC}\left (4 l , -n , n , -4 p l , 2 \left (n +1+p \right ) l -\frac {n^{2}}{2}+m -n , -\frac {x}{2}\right )}\right ] \\ \end{align*}