Link to actual problem [10851] \[ \boxed {y^{\prime \prime }+\left (a x +b \right ) y^{\prime }+\left (c x +d \right ) y=0} \]
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 &= {\mathrm e}^{-\frac {c x}{a}} \operatorname {KummerM}\left (\frac {a^{2} d -a b c +c^{2}}{2 a^{3}}, \frac {1}{2}, -\frac {\left (x \,a^{2}+a b -2 c \right )^{2}}{2 a^{3}}\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{\frac {c x}{a}} y}{\operatorname {KummerM}\left (\frac {a^{2} d -a b c +c^{2}}{2 a^{3}}, \frac {1}{2}, -\frac {\left (x \,a^{2}+a b -2 c \right )^{2}}{2 a^{3}}\right )}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= {\mathrm e}^{-\frac {c x}{a}} \operatorname {KummerU}\left (\frac {a^{2} d -a b c +c^{2}}{2 a^{3}}, \frac {1}{2}, -\frac {\left (x \,a^{2}+a b -2 c \right )^{2}}{2 a^{3}}\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{\frac {c x}{a}} y}{\operatorname {KummerU}\left (\frac {a^{2} d -a b c +c^{2}}{2 a^{3}}, \frac {1}{2}, -\frac {\left (x \,a^{2}+a b -2 c \right )^{2}}{2 a^{3}}\right )}\right ] \\ \end{align*}