Link to actual problem [9472] \[ \boxed {2 a x y^{\prime \prime }+\left (b x +a \right ) y^{\prime }+y c=0} \]
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 &= {\mathrm e}^{-\frac {b x}{2 a}} \operatorname {KummerM}\left (1-\frac {c}{b}, \frac {3}{2}, \frac {b x}{2 a}\right ) \sqrt {x}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{\frac {b x}{2 a}} y}{\operatorname {KummerM}\left (\frac {b -c}{b}, \frac {3}{2}, \frac {b x}{2 a}\right ) \sqrt {x}}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= {\mathrm e}^{-\frac {b x}{2 a}} \operatorname {KummerU}\left (1-\frac {c}{b}, \frac {3}{2}, \frac {b x}{2 a}\right ) \sqrt {x}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{\frac {b x}{2 a}} y}{\operatorname {KummerU}\left (\frac {b -c}{b}, \frac {3}{2}, \frac {b x}{2 a}\right ) \sqrt {x}}\right ] \\ \end{align*}