Link to actual problem [12254] \[ \boxed {x^{2} y^{\prime \prime }-4 x^{2} y^{\prime }+y \left (x^{2}+1\right )=0} \]
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}^{2 x} \sqrt {x}\, \operatorname {BesselI}\left (\frac {i \sqrt {3}}{2}, \sqrt {3}\, x \right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{-2 x} y}{\sqrt {x}\, \operatorname {BesselI}\left (\frac {i \sqrt {3}}{2}, \sqrt {3}\, x \right )}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= {\mathrm e}^{2 x} \sqrt {x}\, \operatorname {BesselK}\left (\frac {i \sqrt {3}}{2}, \sqrt {3}\, x \right )\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{-2 x} y}{\sqrt {x}\, \operatorname {BesselK}\left (\frac {i \sqrt {3}}{2}, \sqrt {3}\, x \right )}\right ] \\ \end{align*}