Link to actual problem [9629] \[ \boxed {\operatorname {A2} \left (x a +b \right )^{2} y^{\prime \prime }+\operatorname {A1} \left (x a +b \right ) y^{\prime }+\operatorname {A0} \left (x a +b \right ) y=0} \]
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 &= \left (x a +b \right )^{\frac {1}{2}-\frac {\operatorname {A1}}{2 a \operatorname {A2}}} \operatorname {BesselJ}\left (1-\frac {\operatorname {A1}}{a \operatorname {A2}}, 2 \sqrt {\operatorname {A0}}\, \sqrt {\frac {x a +b}{a^{2} \operatorname {A2}}}\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {\left (x a +b \right )^{\frac {\operatorname {A1}}{2 a \operatorname {A2}}} y}{\sqrt {x a +b}\, \operatorname {BesselJ}\left (\frac {\operatorname {A2} a -\operatorname {A1}}{a \operatorname {A2}}, 2 \sqrt {\operatorname {A0}}\, \sqrt {\frac {x a +b}{a^{2} \operatorname {A2}}}\right )}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \left (x a +b \right )^{\frac {1}{2}-\frac {\operatorname {A1}}{2 a \operatorname {A2}}} \operatorname {BesselY}\left (1-\frac {\operatorname {A1}}{a \operatorname {A2}}, 2 \sqrt {\operatorname {A0}}\, \sqrt {\frac {x a +b}{a^{2} \operatorname {A2}}}\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {\left (x a +b \right )^{\frac {\operatorname {A1}}{2 a \operatorname {A2}}} y}{\sqrt {x a +b}\, \operatorname {BesselY}\left (\frac {\operatorname {A2} a -\operatorname {A1}}{a \operatorname {A2}}, 2 \sqrt {\operatorname {A0}}\, \sqrt {\frac {x a +b}{a^{2} \operatorname {A2}}}\right )}\right ] \\ \end{align*}