Link to actual problem [10930] \[ \boxed {\left (a x +b \right ) y^{\prime \prime }+s \left (c x +d \right ) y^{\prime }-s^{2} \left (\left (a +c \right ) x +b +d \right ) y=0} \]
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 {s \left (a +c \right ) x}{a}} \operatorname {KummerM}\left (1, 2-\frac {s \left (a d -b c \right )}{a^{2}}, \frac {s \left (2 a +c \right ) \left (x a +b \right )}{a^{2}}\right ) \left (x a +b \right )^{1-\frac {s \left (a d -b c \right )}{a^{2}}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{\frac {s \left (a +c \right ) x}{a}} \left (x a +b \right )^{\frac {s \left (a d -b c \right )}{a^{2}}} y}{\operatorname {KummerM}\left (1, \frac {-a d s +b c s +2 a^{2}}{a^{2}}, \frac {s \left (2 a +c \right ) \left (x a +b \right )}{a^{2}}\right ) \left (x a +b \right )}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= {\mathrm e}^{-\frac {s \left (a +c \right ) x}{a}} \operatorname {KummerU}\left (1, 2-\frac {s \left (a d -b c \right )}{a^{2}}, \frac {s \left (2 a +c \right ) \left (x a +b \right )}{a^{2}}\right ) \left (x a +b \right )^{1-\frac {s \left (a d -b c \right )}{a^{2}}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{\frac {s \left (a +c \right ) x}{a}} \left (x a +b \right )^{\frac {s \left (a d -b c \right )}{a^{2}}} y}{\operatorname {KummerU}\left (1, \frac {-a d s +b c s +2 a^{2}}{a^{2}}, \frac {s \left (2 a +c \right ) \left (x a +b \right )}{a^{2}}\right ) \left (x a +b \right )}\right ] \\ \end{align*}