Link to actual problem [9694] \[ \boxed {y^{\prime \prime }+\frac {2 x y^{\prime }}{x^{2}+1}+\frac {\left (a^{2} \left (x^{2}+1\right )^{2}-n \left (1+n \right ) \left (x^{2}+1\right )+m^{2}\right ) y}{\left (x^{2}+1\right )^{2}}=0} \]
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 (x^{2}+1\right )^{\frac {m}{2}} \operatorname {HeunC}\left (0, -\frac {1}{2}, m , -\frac {a^{2}}{4}, \frac {1}{4}+\frac {1}{4} a^{2}+\frac {1}{4} m^{2}-\frac {1}{4} n^{2}-\frac {1}{4} n , -x^{2}\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {\left (x^{2}+1\right )^{-\frac {m}{2}} y}{\operatorname {HeunC}\left (0, -\frac {1}{2}, m , -\frac {a^{2}}{4}, \frac {1}{4}+\frac {1}{4} a^{2}+\frac {1}{4} m^{2}-\frac {1}{4} n^{2}-\frac {1}{4} n , -x^{2}\right )}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \left (x^{2}+1\right )^{\frac {m}{2}} x \operatorname {HeunC}\left (0, \frac {1}{2}, m , -\frac {a^{2}}{4}, \frac {1}{4}+\frac {1}{4} a^{2}+\frac {1}{4} m^{2}-\frac {1}{4} n^{2}-\frac {1}{4} n , -x^{2}\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {\left (x^{2}+1\right )^{-\frac {m}{2}} y}{x \operatorname {HeunC}\left (0, \frac {1}{2}, m , -\frac {a^{2}}{4}, \frac {1}{4}+\frac {1}{4} a^{2}+\frac {1}{4} m^{2}-\frac {1}{4} n^{2}-\frac {1}{4} n , -x^{2}\right )}\right ] \\ \end{align*}