Link to actual problem [10872] \[ \boxed {y^{\prime \prime }+2 a \,x^{n} y^{\prime }+\left (a^{2} x^{2 n}+b \,x^{2 m}+a n \,x^{n -1}+c \,x^{m -1}\right ) y=0} \]
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 \,{\mathrm e}^{\frac {-i \left (n +1\right ) \sqrt {b}\, x^{1+m}-x^{n +1} a \left (1+m \right )}{\left (n +1\right ) \left (1+m \right )}} \operatorname {KummerM}\left (\frac {i c +\sqrt {b}\, m +2 \sqrt {b}}{2 \sqrt {b}\, \left (1+m \right )}, \frac {m +2}{1+m}, \frac {2 i \sqrt {b}\, x^{1+m}}{1+m}\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{\frac {\left (i \left (n +1\right ) x^{m} \sqrt {b}+a \left (1+m \right ) x^{n}\right ) x}{\left (n +1\right ) \left (1+m \right )}} y}{x \operatorname {KummerM}\left (\frac {\left (m +2\right ) \sqrt {b}+i c}{\sqrt {b}\, \left (2 m +2\right )}, \frac {m +2}{1+m}, \frac {2 i \sqrt {b}\, x \,x^{m}}{1+m}\right )}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= x \,{\mathrm e}^{\frac {-i \left (n +1\right ) \sqrt {b}\, x^{1+m}-x^{n +1} a \left (1+m \right )}{\left (n +1\right ) \left (1+m \right )}} \operatorname {KummerU}\left (\frac {i c +\sqrt {b}\, m +2 \sqrt {b}}{2 \sqrt {b}\, \left (1+m \right )}, \frac {m +2}{1+m}, \frac {2 i \sqrt {b}\, x^{1+m}}{1+m}\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{\frac {\left (i \left (n +1\right ) x^{m} \sqrt {b}+a \left (1+m \right ) x^{n}\right ) x}{\left (n +1\right ) \left (1+m \right )}} y}{x \operatorname {KummerU}\left (\frac {\left (m +2\right ) \sqrt {b}+i c}{\sqrt {b}\, \left (2 m +2\right )}, \frac {m +2}{1+m}, \frac {2 i \sqrt {b}\, x \,x^{m}}{1+m}\right )}\right ] \\ \end{align*}