Link to actual problem [9695] \[ \boxed {y^{\prime \prime }+\frac {a x y^{\prime }}{x^{2}+1}+\frac {b y}{\left (x^{2}+1\right )^{2}}=0} \]
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 {1}{2}-\frac {a}{4}} \operatorname {LegendreP}\left (-1+\frac {a}{2}, \frac {\sqrt {a^{2}-4 a +4 b +4}}{2}, i x \right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {\left (x^{2}+1\right )^{\frac {a}{4}} y}{\sqrt {x^{2}+1}\, \operatorname {LegendreP}\left (-1+\frac {a}{2}, \frac {\sqrt {a^{2}-4 a +4 b +4}}{2}, i x \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 {1}{2}-\frac {a}{4}} \operatorname {LegendreQ}\left (-1+\frac {a}{2}, \frac {\sqrt {a^{2}-4 a +4 b +4}}{2}, i x \right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {\left (x^{2}+1\right )^{\frac {a}{4}} y}{\sqrt {x^{2}+1}\, \operatorname {LegendreQ}\left (-1+\frac {a}{2}, \frac {\sqrt {a^{2}-4 a +4 b +4}}{2}, i x \right )}\right ] \\ \end{align*}