Link to actual problem [11006] \[ \boxed {x^{3} y^{\prime \prime }+\left (a \,x^{2}+b \right ) y^{\prime }+y c x=0} \]
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^{-\frac {\sqrt {a^{2}-2 a -4 c +1}}{2}-\frac {a}{2}+\frac {1}{2}} \operatorname {KummerM}\left (-\frac {1}{4}+\frac {a}{4}+\frac {\sqrt {a^{2}-2 a -4 c +1}}{4}, 1+\frac {\sqrt {a^{2}-2 a -4 c +1}}{2}, \frac {b}{2 x^{2}}\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {x^{\frac {\sqrt {a^{2}-2 a -4 c +1}}{2}} x^{\frac {a}{2}} y}{\sqrt {x}\, \operatorname {KummerM}\left (-\frac {1}{4}+\frac {a}{4}+\frac {\sqrt {a^{2}-2 a -4 c +1}}{4}, 1+\frac {\sqrt {a^{2}-2 a -4 c +1}}{2}, \frac {b}{2 x^{2}}\right )}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= x^{-\frac {\sqrt {a^{2}-2 a -4 c +1}}{2}-\frac {a}{2}+\frac {1}{2}} \operatorname {KummerU}\left (-\frac {1}{4}+\frac {a}{4}+\frac {\sqrt {a^{2}-2 a -4 c +1}}{4}, 1+\frac {\sqrt {a^{2}-2 a -4 c +1}}{2}, \frac {b}{2 x^{2}}\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {x^{\frac {\sqrt {a^{2}-2 a -4 c +1}}{2}} x^{\frac {a}{2}} y}{\sqrt {x}\, \operatorname {KummerU}\left (-\frac {1}{4}+\frac {a}{4}+\frac {\sqrt {a^{2}-2 a -4 c +1}}{4}, 1+\frac {\sqrt {a^{2}-2 a -4 c +1}}{2}, \frac {b}{2 x^{2}}\right )}\right ] \\ \end{align*}