Link to actual problem [11067] \[ \boxed {x^{n} y^{\prime \prime }+\left (a \,x^{n -1}+b x \right ) y^{\prime }+\left (a -1\right ) y=0} \]
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^{-\frac {a}{2}-\frac {1}{2}+\frac {n}{2}} {\mathrm e}^{\frac {b \,x^{-n +2}}{-4+2 n}} \operatorname {WhittakerM}\left (-\frac {a b -b n -2 a +b +2}{2 b \left (n -2\right )}, \frac {a -1}{-4+2 n}, \frac {b \,x^{-n +2}}{n -2}\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {x^{\frac {a}{2}} \sqrt {x}\, x^{-\frac {n}{2}} {\mathrm e}^{-\frac {b \,x^{-n +2}}{2 \left (n -2\right )}} y}{\operatorname {WhittakerM}\left (\frac {\left (-1-a +n \right ) b +2 a -2}{2 b \left (n -2\right )}, \frac {a -1}{-4+2 n}, \frac {x^{-n} b \,x^{2}}{n -2}\right )}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= x^{-\frac {a}{2}-\frac {1}{2}+\frac {n}{2}} {\mathrm e}^{\frac {b \,x^{-n +2}}{-4+2 n}} \operatorname {WhittakerW}\left (-\frac {a b -b n -2 a +b +2}{2 b \left (n -2\right )}, \frac {a -1}{-4+2 n}, \frac {b \,x^{-n +2}}{n -2}\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {x^{\frac {a}{2}} \sqrt {x}\, x^{-\frac {n}{2}} {\mathrm e}^{-\frac {b \,x^{-n +2}}{2 \left (n -2\right )}} y}{\operatorname {WhittakerW}\left (\frac {\left (-1-a +n \right ) b +2 a -2}{2 b \left (n -2\right )}, \frac {a -1}{-4+2 n}, \frac {x^{-n} b \,x^{2}}{n -2}\right )}\right ] \\ \end{align*}