Link to actual problem [10989] \[ \boxed {\left (a \,x^{2}+b \right ) y^{\prime \prime }+\left (1+2 n \right ) a x y^{\prime }+y c=0} \]
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} a +b \right )^{-\frac {n}{2}+\frac {1}{4}} \operatorname {LegendreP}\left (\frac {\sqrt {a \,n^{2}-c}}{\sqrt {a}}-\frac {1}{2}, n -\frac {1}{2}, \frac {a x}{\sqrt {-a b}}\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {\left (x^{2} a +b \right )^{\frac {n}{2}} y}{\left (x^{2} a +b \right )^{\frac {1}{4}} \operatorname {LegendreP}\left (-\frac {-2 \sqrt {a \,n^{2}-c}+\sqrt {a}}{2 \sqrt {a}}, n -\frac {1}{2}, \frac {a x}{\sqrt {-a b}}\right )}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \left (x^{2} a +b \right )^{-\frac {n}{2}+\frac {1}{4}} \operatorname {LegendreQ}\left (\frac {\sqrt {a \,n^{2}-c}}{\sqrt {a}}-\frac {1}{2}, n -\frac {1}{2}, \frac {a x}{\sqrt {-a b}}\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {\left (x^{2} a +b \right )^{\frac {n}{2}} y}{\left (x^{2} a +b \right )^{\frac {1}{4}} \operatorname {LegendreQ}\left (-\frac {-2 \sqrt {a \,n^{2}-c}+\sqrt {a}}{2 \sqrt {a}}, n -\frac {1}{2}, \frac {a x}{\sqrt {-a b}}\right )}\right ] \\ \end{align*}