Link to actual problem [9691] \[ \boxed {y^{\prime \prime }-\frac {\left (2 b c \,x^{c} \left (x^{2}-1\right )+2 \left (-1+a \right ) x^{2}-2 a \right ) y^{\prime }}{x \left (x^{2}-1\right )}+\frac {\left (b^{2} c^{2} x^{2 c} \left (x^{2}-1\right )+b c \,x^{c +2} \left (2 a -c -1\right )-b c \,x^{c} \left (2 a -c +1\right )+x^{2} \left (a \left (-1+a \right )-v \left (v +1\right )\right )-a \left (1+a \right )\right ) y}{x^{2} \left (x^{2}-1\right )}=0} \]
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 [R &= x, S \left (R \right ) &= \frac {x^{-a} {\mathrm e}^{-b \,x^{c}} y}{\operatorname {LegendreP}\left (v , x\right )}\right ] \\ \end{align*}
\begin{align*} \\ \left [R &= x, S \left (R \right ) &= \frac {x^{-a} {\mathrm e}^{-b \,x^{c}} y}{\operatorname {LegendreQ}\left (v , x\right )}\right ] \\ \end{align*}