Link to actual problem [10931] \[ \boxed {\left (a_{2} x +b_{2} \right ) y^{\prime \prime }+\left (a_{1} x +b_{1} \right ) y^{\prime }+\left (a_{0} x +b_{0} \right ) y=0} \]
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 {\left (\sqrt {-4 a_{0} a_{2} +a_{1}^{2}}+a_{1} \right ) x}{2 a_{2}}} \operatorname {KummerM}\left (1+\frac {\sqrt {-4 a_{0} a_{2} +a_{1}^{2}}\, a_{1} b_{2} -\sqrt {-4 a_{0} a_{2} +a_{1}^{2}}\, a_{2} b_{1} +2 a_{2} a_{0} b_{2} -a_{1}^{2} b_{2} +a_{1} a_{2} b_{1} -2 a_{2}^{2} b_{0}}{2 a_{2}^{2} \sqrt {-4 a_{0} a_{2} +a_{1}^{2}}}, 2+\frac {a_{1} b_{2} -a_{2} b_{1}}{a_{2}^{2}}, \frac {\sqrt {-4 a_{0} a_{2} +a_{1}^{2}}\, \left (a_{2} x +b_{2} \right )}{a_{2}^{2}}\right ) \left (a_{2} x +b_{2} \right )^{1+\frac {a_{1} b_{2} -a_{2} b_{1}}{a_{2}^{2}}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{\frac {\left (\sqrt {-4 a_{0} a_{2} +a_{1}^{2}}+a_{1} \right ) x}{2 a_{2}}} \left (a_{2} x +b_{2} \right )^{-\frac {a_{1} b_{2} -a_{2} b_{1}}{a_{2}^{2}}} y}{\operatorname {KummerM}\left (\frac {\left (a_{1} b_{2} +2 a_{2}^{2}-a_{2} b_{1} \right ) \sqrt {-4 a_{0} a_{2} +a_{1}^{2}}-2 a_{2}^{2} b_{0} +\left (2 a_{0} b_{2} +a_{1} b_{1} \right ) a_{2} -a_{1}^{2} b_{2}}{2 \sqrt {-4 a_{0} a_{2} +a_{1}^{2}}\, a_{2}^{2}}, \frac {a_{1} b_{2} +2 a_{2}^{2}-a_{2} b_{1}}{a_{2}^{2}}, \frac {\sqrt {-4 a_{0} a_{2} +a_{1}^{2}}\, \left (a_{2} x +b_{2} \right )}{a_{2}^{2}}\right ) \left (a_{2} x +b_{2} \right )}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= {\mathrm e}^{-\frac {\left (\sqrt {-4 a_{0} a_{2} +a_{1}^{2}}+a_{1} \right ) x}{2 a_{2}}} \operatorname {KummerU}\left (1+\frac {\sqrt {-4 a_{0} a_{2} +a_{1}^{2}}\, a_{1} b_{2} -\sqrt {-4 a_{0} a_{2} +a_{1}^{2}}\, a_{2} b_{1} +2 a_{2} a_{0} b_{2} -a_{1}^{2} b_{2} +a_{1} a_{2} b_{1} -2 a_{2}^{2} b_{0}}{2 a_{2}^{2} \sqrt {-4 a_{0} a_{2} +a_{1}^{2}}}, 2+\frac {a_{1} b_{2} -a_{2} b_{1}}{a_{2}^{2}}, \frac {\sqrt {-4 a_{0} a_{2} +a_{1}^{2}}\, \left (a_{2} x +b_{2} \right )}{a_{2}^{2}}\right ) \left (a_{2} x +b_{2} \right )^{1+\frac {a_{1} b_{2} -a_{2} b_{1}}{a_{2}^{2}}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{\frac {\left (\sqrt {-4 a_{0} a_{2} +a_{1}^{2}}+a_{1} \right ) x}{2 a_{2}}} \left (a_{2} x +b_{2} \right )^{-\frac {a_{1} b_{2} -a_{2} b_{1}}{a_{2}^{2}}} y}{\operatorname {KummerU}\left (\frac {\left (a_{1} b_{2} +2 a_{2}^{2}-a_{2} b_{1} \right ) \sqrt {-4 a_{0} a_{2} +a_{1}^{2}}-2 a_{2}^{2} b_{0} +\left (2 a_{0} b_{2} +a_{1} b_{1} \right ) a_{2} -a_{1}^{2} b_{2}}{2 \sqrt {-4 a_{0} a_{2} +a_{1}^{2}}\, a_{2}^{2}}, \frac {a_{1} b_{2} +2 a_{2}^{2}-a_{2} b_{1}}{a_{2}^{2}}, \frac {\sqrt {-4 a_{0} a_{2} +a_{1}^{2}}\, \left (a_{2} x +b_{2} \right )}{a_{2}^{2}}\right ) \left (a_{2} x +b_{2} \right )}\right ] \\ \end{align*}