Link to actual problem [12261] \[ \boxed {x^{2} y^{\prime \prime }+x^{2} y^{\prime }+2 \left (1-x \right ) y=0} \]
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 {x}{2}} \sqrt {x}\, \left (\left (x^{2}+2 x \right ) \operatorname {BesselI}\left (\frac {i \sqrt {7}}{2}+1, \frac {x}{2}\right )+\operatorname {BesselI}\left (\frac {i \sqrt {7}}{2}, \frac {x}{2}\right ) \left (-2+i \left (2+x \right ) \sqrt {7}+x^{2}+3 x \right )\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{\frac {x}{2}} y}{\sqrt {x}\, \left (\left (x^{2}+2 x \right ) \operatorname {BesselI}\left (\frac {i \sqrt {7}}{2}+1, \frac {x}{2}\right )+\operatorname {BesselI}\left (\frac {i \sqrt {7}}{2}, \frac {x}{2}\right ) \left (-2+i \left (2+x \right ) \sqrt {7}+x^{2}+3 x \right )\right )}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= {\mathrm e}^{-\frac {x}{2}} \sqrt {x}\, \left (\left (-x^{2}-2 x \right ) \operatorname {BesselK}\left (\frac {i \sqrt {7}}{2}+1, \frac {x}{2}\right )+\operatorname {BesselK}\left (\frac {i \sqrt {7}}{2}, \frac {x}{2}\right ) \left (-2+i \left (2+x \right ) \sqrt {7}+x^{2}+3 x \right )\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{\frac {x}{2}} y}{\sqrt {x}\, \left (\left (-x^{2}-2 x \right ) \operatorname {BesselK}\left (\frac {i \sqrt {7}}{2}+1, \frac {x}{2}\right )+\operatorname {BesselK}\left (\frac {i \sqrt {7}}{2}, \frac {x}{2}\right ) \left (-2+i \left (2+x \right ) \sqrt {7}+x^{2}+3 x \right )\right )}\right ] \\ \end{align*}