Link to actual problem [10870] \[ \boxed {y^{\prime \prime }+a \,x^{n} y^{\prime }+\left (b \,x^{2 n}+c \,x^{n -1}\right ) y=0} \]
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 &= x \,{\mathrm e}^{-\frac {x^{n +1} \left (a +\sqrt {a^{2}-4 b}\right )}{2 \left (n +1\right )}} \operatorname {KummerM}\left (\frac {\sqrt {a^{2}-4 b}\, n +a n +2 \sqrt {a^{2}-4 b}-2 c}{2 \sqrt {a^{2}-4 b}\, \left (n +1\right )}, \frac {n +2}{n +1}, \frac {\sqrt {a^{2}-4 b}\, x^{n +1}}{n +1}\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{\frac {x \,x^{n} \left (a +\sqrt {a^{2}-4 b}\right )}{2 n +2}} y}{x \operatorname {KummerM}\left (\frac {\left (n +2\right ) \sqrt {a^{2}-4 b}+a n -2 c}{\sqrt {a^{2}-4 b}\, \left (2 n +2\right )}, \frac {n +2}{n +1}, \frac {\sqrt {a^{2}-4 b}\, x \,x^{n}}{n +1}\right )}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= x \,{\mathrm e}^{-\frac {x^{n +1} \left (a +\sqrt {a^{2}-4 b}\right )}{2 \left (n +1\right )}} \operatorname {KummerU}\left (\frac {\sqrt {a^{2}-4 b}\, n +a n +2 \sqrt {a^{2}-4 b}-2 c}{2 \sqrt {a^{2}-4 b}\, \left (n +1\right )}, \frac {n +2}{n +1}, \frac {\sqrt {a^{2}-4 b}\, x^{n +1}}{n +1}\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{\frac {x \,x^{n} \left (a +\sqrt {a^{2}-4 b}\right )}{2 n +2}} y}{x \operatorname {KummerU}\left (\frac {\left (n +2\right ) \sqrt {a^{2}-4 b}+a n -2 c}{\sqrt {a^{2}-4 b}\, \left (2 n +2\right )}, \frac {n +2}{n +1}, \frac {\sqrt {a^{2}-4 b}\, x \,x^{n}}{n +1}\right )}\right ] \\ \end{align*}