Link to actual problem [11008] \[ \boxed {x^{3} y^{\prime \prime }+\left (a \,x^{2}+b x \right ) y^{\prime }+y c=0} \]
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^{1-a} \operatorname {KummerM}\left (a -1-\frac {c}{b}, a , \frac {b}{x}\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {x^{a} y}{x \operatorname {KummerM}\left (\frac {b \left (a -1\right )-c}{b}, a , \frac {b}{x}\right )}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= x^{1-a} \operatorname {KummerU}\left (a -1-\frac {c}{b}, a , \frac {b}{x}\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {x^{a} y}{x \operatorname {KummerU}\left (\frac {b \left (a -1\right )-c}{b}, a , \frac {b}{x}\right )}\right ] \\ \end{align*}