Link to actual problem [9535] \[ \boxed {x^{2} y^{\prime \prime }+\left (2 x a +b \right ) x y^{\prime }+\left (a b x +c \,x^{2}+d \right ) y=0} \]
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 &= x^{\frac {1}{2}-\frac {b}{2}} {\mathrm e}^{-x a} \operatorname {BesselJ}\left (\frac {\sqrt {b^{2}-2 b -4 d +1}}{2}, \sqrt {-a^{2}+c}\, x \right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {x^{\frac {b}{2}} {\mathrm e}^{x a} y}{\sqrt {x}\, \operatorname {BesselJ}\left (\frac {\sqrt {b^{2}-2 b -4 d +1}}{2}, \sqrt {-a^{2}+c}\, x \right )}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= x^{\frac {1}{2}-\frac {b}{2}} {\mathrm e}^{-x a} \operatorname {BesselY}\left (\frac {\sqrt {b^{2}-2 b -4 d +1}}{2}, \sqrt {-a^{2}+c}\, x \right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {x^{\frac {b}{2}} {\mathrm e}^{x a} y}{\sqrt {x}\, \operatorname {BesselY}\left (\frac {\sqrt {b^{2}-2 b -4 d +1}}{2}, \sqrt {-a^{2}+c}\, x \right )}\right ] \\ \end{align*}