Link to actual problem [9474] \[ \boxed {\left (\operatorname {a2} x +\operatorname {b2} \right ) y^{\prime \prime }+\left (\operatorname {a1} x +\operatorname {b1} \right ) y^{\prime }+\left (\operatorname {a0} x +\operatorname {b0} \right ) y=0} \]
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 \operatorname {a0} \operatorname {a2} +\operatorname {a1}^{2}}+\operatorname {a1} \right ) x}{2 \operatorname {a2}}} \operatorname {KummerM}\left (1+\frac {\sqrt {-4 \operatorname {a0} \operatorname {a2} +\operatorname {a1}^{2}}\, \operatorname {a1} \operatorname {b2} -\sqrt {-4 \operatorname {a0} \operatorname {a2} +\operatorname {a1}^{2}}\, \operatorname {a2} \operatorname {b1} +2 \operatorname {a2} \operatorname {a0} \operatorname {b2} -\operatorname {a1}^{2} \operatorname {b2} +\operatorname {a1} \operatorname {a2} \operatorname {b1} -2 \operatorname {a2}^{2} \operatorname {b0}}{2 \operatorname {a2}^{2} \sqrt {-4 \operatorname {a0} \operatorname {a2} +\operatorname {a1}^{2}}}, 2+\frac {\operatorname {a1} \operatorname {b2} -\operatorname {a2} \operatorname {b1}}{\operatorname {a2}^{2}}, \frac {\sqrt {-4 \operatorname {a0} \operatorname {a2} +\operatorname {a1}^{2}}\, \left (\operatorname {a2} x +\operatorname {b2} \right )}{\operatorname {a2}^{2}}\right ) \left (\operatorname {a2} x +\operatorname {b2} \right )^{1+\frac {\operatorname {a1} \operatorname {b2} -\operatorname {a2} \operatorname {b1}}{\operatorname {a2}^{2}}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{\frac {\left (\sqrt {-4 \operatorname {a0} \operatorname {a2} +\operatorname {a1}^{2}}+\operatorname {a1} \right ) x}{2 \operatorname {a2}}} \left (\operatorname {a2} x +\operatorname {b2} \right )^{-\frac {\operatorname {a1} \operatorname {b2} -\operatorname {a2} \operatorname {b1}}{\operatorname {a2}^{2}}} y}{\operatorname {KummerM}\left (\frac {\left (\operatorname {a1} \operatorname {b2} +2 \operatorname {a2}^{2}-\operatorname {a2} \operatorname {b1} \right ) \sqrt {-4 \operatorname {a0} \operatorname {a2} +\operatorname {a1}^{2}}-2 \operatorname {a2}^{2} \operatorname {b0} +\left (2 \operatorname {a0} \operatorname {b2} +\operatorname {a1} \operatorname {b1} \right ) \operatorname {a2} -\operatorname {a1}^{2} \operatorname {b2}}{2 \sqrt {-4 \operatorname {a0} \operatorname {a2} +\operatorname {a1}^{2}}\, \operatorname {a2}^{2}}, \frac {\operatorname {a1} \operatorname {b2} +2 \operatorname {a2}^{2}-\operatorname {a2} \operatorname {b1}}{\operatorname {a2}^{2}}, \frac {\sqrt {-4 \operatorname {a0} \operatorname {a2} +\operatorname {a1}^{2}}\, \left (\operatorname {a2} x +\operatorname {b2} \right )}{\operatorname {a2}^{2}}\right ) \left (\operatorname {a2} x +\operatorname {b2} \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 \operatorname {a0} \operatorname {a2} +\operatorname {a1}^{2}}+\operatorname {a1} \right ) x}{2 \operatorname {a2}}} \operatorname {KummerU}\left (1+\frac {\sqrt {-4 \operatorname {a0} \operatorname {a2} +\operatorname {a1}^{2}}\, \operatorname {a1} \operatorname {b2} -\sqrt {-4 \operatorname {a0} \operatorname {a2} +\operatorname {a1}^{2}}\, \operatorname {a2} \operatorname {b1} +2 \operatorname {a2} \operatorname {a0} \operatorname {b2} -\operatorname {a1}^{2} \operatorname {b2} +\operatorname {a1} \operatorname {a2} \operatorname {b1} -2 \operatorname {a2}^{2} \operatorname {b0}}{2 \operatorname {a2}^{2} \sqrt {-4 \operatorname {a0} \operatorname {a2} +\operatorname {a1}^{2}}}, 2+\frac {\operatorname {a1} \operatorname {b2} -\operatorname {a2} \operatorname {b1}}{\operatorname {a2}^{2}}, \frac {\sqrt {-4 \operatorname {a0} \operatorname {a2} +\operatorname {a1}^{2}}\, \left (\operatorname {a2} x +\operatorname {b2} \right )}{\operatorname {a2}^{2}}\right ) \left (\operatorname {a2} x +\operatorname {b2} \right )^{1+\frac {\operatorname {a1} \operatorname {b2} -\operatorname {a2} \operatorname {b1}}{\operatorname {a2}^{2}}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{\frac {\left (\sqrt {-4 \operatorname {a0} \operatorname {a2} +\operatorname {a1}^{2}}+\operatorname {a1} \right ) x}{2 \operatorname {a2}}} \left (\operatorname {a2} x +\operatorname {b2} \right )^{-\frac {\operatorname {a1} \operatorname {b2} -\operatorname {a2} \operatorname {b1}}{\operatorname {a2}^{2}}} y}{\operatorname {KummerU}\left (\frac {\left (\operatorname {a1} \operatorname {b2} +2 \operatorname {a2}^{2}-\operatorname {a2} \operatorname {b1} \right ) \sqrt {-4 \operatorname {a0} \operatorname {a2} +\operatorname {a1}^{2}}-2 \operatorname {a2}^{2} \operatorname {b0} +\left (2 \operatorname {a0} \operatorname {b2} +\operatorname {a1} \operatorname {b1} \right ) \operatorname {a2} -\operatorname {a1}^{2} \operatorname {b2}}{2 \sqrt {-4 \operatorname {a0} \operatorname {a2} +\operatorname {a1}^{2}}\, \operatorname {a2}^{2}}, \frac {\operatorname {a1} \operatorname {b2} +2 \operatorname {a2}^{2}-\operatorname {a2} \operatorname {b1}}{\operatorname {a2}^{2}}, \frac {\sqrt {-4 \operatorname {a0} \operatorname {a2} +\operatorname {a1}^{2}}\, \left (\operatorname {a2} x +\operatorname {b2} \right )}{\operatorname {a2}^{2}}\right ) \left (\operatorname {a2} x +\operatorname {b2} \right )}\right ] \\ \end{align*}