Link to actual problem [10922] \[ \boxed {x y^{\prime \prime }+\left (a \,x^{n}+b \right ) y^{\prime }+\left (x^{2 n -1} c +d \,x^{n -1}\right ) y=0} \]
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 &= {\mathrm e}^{-\frac {x^{n} \left (\sqrt {a^{2}-4 c}+a \right )}{2 n}} \operatorname {KummerM}\left (\frac {1}{2}+\frac {b \sqrt {a^{2}-4 c}+a b +a n -\sqrt {a^{2}-4 c}-a -2 d}{2 \sqrt {a^{2}-4 c}\, n}, 1+\frac {b -1}{n}, \frac {\sqrt {a^{2}-4 c}\, x^{n}}{n}\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{\frac {x^{n} \left (\sqrt {a^{2}-4 c}+a \right )}{2 n}} y}{\operatorname {KummerM}\left (\frac {\left (b +n -1\right ) \sqrt {a^{2}-4 c}+\left (b +n -1\right ) a -2 d}{2 \sqrt {a^{2}-4 c}\, n}, \frac {b +n -1}{n}, \frac {\sqrt {a^{2}-4 c}\, x^{n}}{n}\right )}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= {\mathrm e}^{-\frac {x^{n} \left (\sqrt {a^{2}-4 c}+a \right )}{2 n}} \operatorname {KummerU}\left (\frac {1}{2}+\frac {b \sqrt {a^{2}-4 c}+a b +a n -\sqrt {a^{2}-4 c}-a -2 d}{2 \sqrt {a^{2}-4 c}\, n}, 1+\frac {b -1}{n}, \frac {\sqrt {a^{2}-4 c}\, x^{n}}{n}\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{\frac {x^{n} \left (\sqrt {a^{2}-4 c}+a \right )}{2 n}} y}{\operatorname {KummerU}\left (\frac {\left (b +n -1\right ) \sqrt {a^{2}-4 c}+\left (b +n -1\right ) a -2 d}{2 \sqrt {a^{2}-4 c}\, n}, \frac {b +n -1}{n}, \frac {\sqrt {a^{2}-4 c}\, x^{n}}{n}\right )}\right ] \\ \end{align*}