Link to actual problem [10929] \[ \boxed {\left (a_{1} x +a_{0} \right ) y^{\prime \prime }+\left (b_{1} x +b_{0} \right ) y^{\prime }-m b_{1} y=0} \]
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 {b_{1} x}{a_{1}}} \operatorname {KummerM}\left (1+m , 2+\frac {a_{0} b_{1} -a_{1} b_{0}}{a_{1}^{2}}, \frac {b_{1} \left (a_{1} x +a_{0} \right )}{a_{1}^{2}}\right ) \left (a_{1} x +a_{0} \right )^{1+\frac {a_{0} b_{1} -a_{1} b_{0}}{a_{1}^{2}}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{\frac {b_{1} x}{a_{1}}} \left (a_{1} x +a_{0} \right )^{-\frac {a_{0} b_{1} -a_{1} b_{0}}{a_{1}^{2}}} y}{\operatorname {KummerM}\left (1+m , \frac {a_{0} b_{1} +2 a_{1}^{2}-a_{1} b_{0}}{a_{1}^{2}}, \frac {b_{1} \left (a_{1} x +a_{0} \right )}{a_{1}^{2}}\right ) \left (a_{1} x +a_{0} \right )}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= {\mathrm e}^{-\frac {b_{1} x}{a_{1}}} \operatorname {KummerU}\left (1+m , 2+\frac {a_{0} b_{1} -a_{1} b_{0}}{a_{1}^{2}}, \frac {b_{1} \left (a_{1} x +a_{0} \right )}{a_{1}^{2}}\right ) \left (a_{1} x +a_{0} \right )^{1+\frac {a_{0} b_{1} -a_{1} b_{0}}{a_{1}^{2}}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{\frac {b_{1} x}{a_{1}}} \left (a_{1} x +a_{0} \right )^{-\frac {a_{0} b_{1} -a_{1} b_{0}}{a_{1}^{2}}} y}{\operatorname {KummerU}\left (1+m , \frac {a_{0} b_{1} +2 a_{1}^{2}-a_{1} b_{0}}{a_{1}^{2}}, \frac {b_{1} \left (a_{1} x +a_{0} \right )}{a_{1}^{2}}\right ) \left (a_{1} x +a_{0} \right )}\right ] \\ \end{align*}