Link to actual problem [10868] \[ \boxed {y^{\prime \prime }+a \,x^{n} y^{\prime }+y x^{n -1} b=0} \]
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 {a \,x^{n +1}}{n +1}} \operatorname {KummerM}\left (\frac {a n +a -b}{a \left (n +1\right )}, \frac {n +2}{n +1}, \frac {a \,x^{n +1}}{n +1}\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{\frac {a x \,x^{n}}{n +1}} y}{x \operatorname {KummerM}\left (\frac {a \left (n +1\right )-b}{a \left (n +1\right )}, \frac {n +2}{n +1}, \frac {a 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 {a \,x^{n +1}}{n +1}} \operatorname {KummerU}\left (\frac {a n +a -b}{a \left (n +1\right )}, \frac {n +2}{n +1}, \frac {a \,x^{n +1}}{n +1}\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{\frac {a x \,x^{n}}{n +1}} y}{x \operatorname {KummerU}\left (\frac {a \left (n +1\right )-b}{a \left (n +1\right )}, \frac {n +2}{n +1}, \frac {a x \,x^{n}}{n +1}\right )}\right ] \\ \end{align*}