Link to actual problem [9524] \[ \boxed {x^{2} y^{\prime \prime }+\left (x +3\right ) x y^{\prime }-y=0} \]
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 &= \frac {{\mathrm e}^{-\frac {x}{2}} \left (\left (\sqrt {2}+x +1\right ) \operatorname {BesselI}\left (-\frac {1}{2}+\sqrt {2}, \frac {x}{2}\right )+\left (-\sqrt {2}+x +1\right ) \operatorname {BesselI}\left (\frac {1}{2}+\sqrt {2}, \frac {x}{2}\right )\right )}{\sqrt {x}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{\frac {x}{2}} \sqrt {x}\, y}{\left (\sqrt {2}+x +1\right ) \operatorname {BesselI}\left (-\frac {1}{2}+\sqrt {2}, \frac {x}{2}\right )+\left (-\sqrt {2}+x +1\right ) \operatorname {BesselI}\left (\frac {1}{2}+\sqrt {2}, \frac {x}{2}\right )}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \frac {{\mathrm e}^{-\frac {x}{2}} \left (\left (\sqrt {2}+x +1\right ) \operatorname {BesselK}\left (-\frac {1}{2}+\sqrt {2}, \frac {x}{2}\right )-\operatorname {BesselK}\left (\frac {1}{2}+\sqrt {2}, \frac {x}{2}\right ) \left (-\sqrt {2}+x +1\right )\right )}{\sqrt {x}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {\sqrt {x}\, {\mathrm e}^{\frac {x}{2}} y}{\left (\sqrt {2}+x +1\right ) \operatorname {BesselK}\left (-\frac {1}{2}+\sqrt {2}, \frac {x}{2}\right )-\operatorname {BesselK}\left (\frac {1}{2}+\sqrt {2}, \frac {x}{2}\right ) \left (-\sqrt {2}+x +1\right )}\right ] \\ \end{align*}