Link to actual problem [7181] \[ \boxed {y^{\prime \prime }-y^{\prime } x^{2}-y x=x^{2}} \]
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*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= {\mathrm e}^{\frac {x^{3}}{6}} \sqrt {x}\, \operatorname {BesselI}\left (\frac {1}{6}, \frac {x^{3}}{6}\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{-\frac {x^{3}}{6}} y}{\sqrt {x}\, \operatorname {BesselI}\left (\frac {1}{6}, \frac {x^{3}}{6}\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}} \sqrt {x}\, \operatorname {BesselK}\left (\frac {1}{6}, \frac {x^{3}}{6}\right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{-\frac {x^{3}}{6}} y}{\sqrt {x}\, \operatorname {BesselK}\left (\frac {1}{6}, \frac {x^{3}}{6}\right )}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \frac {x}{2}+y\right ] \\ \\ \end{align*}