Link to actual problem [7894] \[ \boxed {3 y^{\prime \prime }+x y^{\prime }-4 y=0} \]
type detected by program
{"kovacic"}
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 [R &= x, S \left (R \right ) &= \frac {y}{x^{4}+18 x^{2}+27}\right ] \\ \end{align*}
\begin{align*} \left [\underline {\hspace {1.25 ex}}\xi &= 0, \underline {\hspace {1.25 ex}}\eta &= \sqrt {\pi }\, \operatorname {erf}\left (\frac {\sqrt {6}\, x}{6}\right ) \left (x^{4}+18 x^{2}+27\right )+\left (\sqrt {6}\, x^{3}+15 \sqrt {6}\, x \right ) {\mathrm e}^{-\frac {x^{2}}{6}}\right ] \\ \left [R &= x, S \left (R \right ) &= \frac {{\mathrm e}^{\frac {x^{2}}{6}} y}{\sqrt {\pi }\, \operatorname {erf}\left (\frac {\sqrt {6}\, x}{6}\right ) \left (x^{4}+18 x^{2}+27\right ) {\mathrm e}^{\frac {x^{2}}{6}}+\sqrt {6}\, x \left (x^{2}+15\right )}\right ] \\ \end{align*}