Link to actual problem [9542] \[ \boxed {x^{2} y^{\prime \prime }+\left (x^{3}+1\right ) x y^{\prime }-y=0} \]
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 &= {\mathrm e}^{-\frac {x^{3}}{6}} x^{\frac {3}{2}} \left (\operatorname {BesselI}\left (\frac {5}{6}, \frac {x^{3}}{6}\right )+\operatorname {BesselI}\left (-\frac {1}{6}, \frac {x^{3}}{6}\right )\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{\frac {x^{3}}{6}} y}{x^{\frac {3}{2}} \left (\operatorname {BesselI}\left (\frac {5}{6}, \frac {x^{3}}{6}\right )+\operatorname {BesselI}\left (-\frac {1}{6}, \frac {x^{3}}{6}\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^{3}}{6}} x^{\frac {3}{2}} \left (\operatorname {BesselK}\left (\frac {1}{6}, \frac {x^{3}}{6}\right )-\operatorname {BesselK}\left (\frac {5}{6}, \frac {x^{3}}{6}\right )\right )\right ] \\ \left [R &= x, S \left (R \right ) &= -\frac {{\mathrm e}^{\frac {x^{3}}{6}} y}{x^{\frac {3}{2}} \left (\operatorname {BesselK}\left (\frac {1}{6}, \frac {x^{3}}{6}\right )-\operatorname {BesselK}\left (\frac {5}{6}, \frac {x^{3}}{6}\right )\right )}\right ] \\ \end{align*}