Link to actual problem [11009] \[ \boxed {x^{3} y^{\prime \prime }+\left (a \,x^{2}+b x \right ) y^{\prime }+\left (c x +d \right ) y=0} \]
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^{-\frac {\sqrt {a^{2}-2 a -4 c +1}}{2}-\frac {a}{2}+\frac {1}{2}} \operatorname {KummerM}\left (-\frac {1}{2}+\frac {\sqrt {a^{2}-2 a -4 c +1}\, b +a b -2 d}{2 b}, 1+\sqrt {a^{2}-2 a -4 c +1}, \frac {b}{x}\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 {\sqrt {a^{2}-2 a -4 c +1}\, b +b \left (a -1\right )-2 d}{2 b}, 1+\sqrt {a^{2}-2 a -4 c +1}, \frac {b}{x}\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}{2}+\frac {\sqrt {a^{2}-2 a -4 c +1}\, b +a b -2 d}{2 b}, 1+\sqrt {a^{2}-2 a -4 c +1}, \frac {b}{x}\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 {\sqrt {a^{2}-2 a -4 c +1}\, b +b \left (a -1\right )-2 d}{2 b}, 1+\sqrt {a^{2}-2 a -4 c +1}, \frac {b}{x}\right )}\right ] \\ \end{align*}