Link to actual problem [9684] \[ \boxed {y^{\prime \prime }+\frac {\left (x^{2} a +a -1\right ) y^{\prime }}{x \left (x^{2}+1\right )}+\frac {\left (b \,x^{2}+c \right ) y}{x^{2} \left (x^{2}+1\right )}=0} \]
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^{1-\frac {a}{2}} \operatorname {LegendreP}\left (-\frac {1}{2}+\frac {\sqrt {a^{2}-2 a -4 b +1}}{2}, \frac {\sqrt {a^{2}-4 a -4 c +4}}{2}, \sqrt {x^{2}+1}\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {x^{\frac {a}{2}} y}{x \operatorname {LegendreP}\left (-\frac {1}{2}+\frac {\sqrt {a^{2}-2 a -4 b +1}}{2}, \frac {\sqrt {a^{2}-4 a -4 c +4}}{2}, \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^{1-\frac {a}{2}} \operatorname {LegendreQ}\left (-\frac {1}{2}+\frac {\sqrt {a^{2}-2 a -4 b +1}}{2}, \frac {\sqrt {a^{2}-4 a -4 c +4}}{2}, \sqrt {x^{2}+1}\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {x^{\frac {a}{2}} y}{x \operatorname {LegendreQ}\left (-\frac {1}{2}+\frac {\sqrt {a^{2}-2 a -4 b +1}}{2}, \frac {\sqrt {a^{2}-4 a -4 c +4}}{2}, \sqrt {x^{2}+1}\right )}\right ] \\ \end{align*}